Department of Physics - LABORATORY HANDBOOK- 2004-2005

7/31/2019 Department of Physics - LABORATORY HANDBOOK- 2004-2005

http://slidepdf.com/reader/full/department-of-physics-laboratory-handbook-2004-2005 1/39

Department of Physics

Year 1

Laboratory

Handbook

2004–2005



First Year Laboratory Handbook – 2 – JHS 18 September, 2004

Department of PhysicsLOUGHBOROUGH UNIVERSITY

YEAR 1 LABORATORY HANDBOOK

CONTENTS

1. L ABORATORYREGULATIONS.......................................................................................................... 42. WHAT YOU SHOULD GET OUT OF THE FIRST YEAR LABORATORY...................................................... 5

2.1 Aims...................................................................................................................................... 5 2.2 Intended Learning Outcomes .............................................................................................. 5

Knowledge and Understanding....................................................................................................5Skills and Attributes ..................................................................................................................... 5

2.3 How to go about it ................................................................................................................ 5

3. YOUR PART IN THE LABORATORY.................................................................................................... 73.1 General................................................................................................................................. 7 3.2 The Assessment Process.................................................................................................... 8 3.3 Deadlines ............................................................................................................................. 8 3.4 In Summary.......................................................................................................................... 93.5 Deadline summary for semesters 1 and 2 (semester 2 provisional) .................................. 93.6 Next year.............................................................................................................................. 9

4. EXPERIMENTALPROCEDURE........................................................................................................ 104.1 On arrival............................................................................................................................ 10 4.2 The preliminary experiment ............................................................................................... 10 4.3 Repetition of measurements.............................................................................................. 10 4.4 Recording information........................................................................................................ 10 4.5 Marking............................................................................................................................... 114.6 Before you leave................................................................................................................ 11

5. L ABORATORY NOTEBOOKS........................................................................................................... 125.1 Errors.................................................................................................................................. 12 5.2 Graphs................................................................................................................................ 12 5.3 Units ................................................................................................................................... 12 5.4 Marking............................................................................................................................... 135.5 Plagiarism........................................................................................................................... 13

6. REPORTS .....................................................................................................................................146.1 Aim of the report................................................................................................................. 146.2 Format of report ................................................................................................................. 14

6.2.1 Language and style........................................................................................................ 146.2.2 Title Page....................................................................................................................... 15

6.2.3 Abstract .......................................................................................................................... 156.2.4 Introduction..................................................................................................................... 156.2.5 Experimental Method..................................................................................................... 156.2.6 Results and Discussion.................................................................................................. 166.2.7 Figures and tables.......................................................................................................... 166.2.8 Conclusions....................................................................................................................166.2.9 References..................................................................................................................... 16

6.3 Assessment........................................................................................................................ 17 7. MEASUREMENTERROR................................................................................................................18

7.1 Introduction......................................................................................................................... 18 7.2 Drawing conclusions from experiments............................................................................. 18 7.3 Presenting a measurement................................................................................................ 197.4 Random and Systematic Errors......................................................................................... 197.5 Adding Errors ..................................................................................................................... 237.6 Error in a function of a measured quantity ........................................................................ 237.7 Error in a function of more than one measured quantity................................................... 25




7.8 Labour Saving Formulae.................................................................................................... 25 8. GRAPHS ......................................................................................................................................27

8.1 Types of graph ................................................................................................................... 27 8.2 Logarithmic Graph Paper................................................................................................... 298.3 Axes.................................................................................................................................... 298.4 Error Bars........................................................................................................................... 30 8.5 Errors in Gradients............................................................................................................. 30

9. UNITS.......................................................................................................................................... 329.1 Use of units ........................................................................................................................ 32 9.2 Basic and supplementary SI units ..................................................................................... 339.3 Derived units ...................................................................................................................... 34

10. S AFETY........................................................................................................................................3510.1 Fire ..................................................................................................................................... 36 10.2 General Laboratory Safety................................................................................................. 36 10.3 Electricity............................................................................................................................ 37 10.4 Lasers................................................................................................................................. 37

REFERENCES AND FURTHER READING.................................................................................................. 38 APPENDIX: S AMPLEREPORT................................................................................................................39




1. Laboratory Regulations

Attendance

There is an Attendance List, which must be signed by each student at the be-ginning of every laboratory session. The times of arriving in and leaving thelaboratory must be entered on the Attendance List.Students are permitted to leave at any time for a tea break, etc., but must in-dicate their absence by recording on the Attendance List the times of their leaving and returning to the laboratory.

SMOKING, EATING, DRINKING, PERSONAL STEREOS AND MOBILETELEPHONES ARE NOT PERMITTED IN THE LABORATORY.

BreakagesBreakages should be reported to a Demonstrator as soon as they occur. Noattempt should be made to repair apparatus before the event has been re-ported.

Injury Any injury, however trivial, must be reported to the laboratory supervisor.This fulfils legal requirements. A First Aid Kit is available.

Reference BooksReference books are kept in the laboratory. These may be borrowed for usewhilst an experiment is in progress. They must not be removed from thelaboratory.

ApparatusEquipment must not be removed from the laboratory without permission.

Safety

The Safety Precautions must be observed at all times. You must read and understand Section 10 (Safety) before starting your first experiment.




2. What you should get out of the first year laboratory

2.1 Aims

1. To gain experience and skills in carrying out experimental work.

2. To illustrate principles in physics.

2.2 Intended Learning Outcomes

These are the knowledge and skills you should gain or further develop through the lab classes, andwhat we are looking for when assessing your work. These differ somewhat between the laboratorymodules.

Knowledge and Understanding

1. How basic principles of physics can be demonstrated by simple experiments2. Knowledge of experimental techniques3. Understanding of the importance of error analysis

Skills and Attributes

(i) Intellectual 1. Understanding of the practical aspects of the physics phenomena described in lectures in

other modules2. Critical awareness and a practical appreciation of phenomena covered in lectures3. Ability to extract maximum information from data

(ii) Practical 1. Understanding how the laws of physics can be applied to practical situations2. Application of the scientific method3. Use of a wide variety of equipment in the physics laboratory4. Keeping complete, accurate and legible records in a log book

(iii) Transferable1. Confidence and facility in systematic approaches to experimental techniques2. Report writing and oral communication skills3. Ability to work either independently or in a team4. Self-organisation to meet deadlines

You will therefore get more out of the laboratory classes than experimental techniques and an un-

derstanding of the physical processes observed; the skills obtained are transferable to many other walks of life.

2.3 How to go about it

You can develop your analytical skill by asking yourself the initial question, “What is it that we aretrying to (have been instructed to) do?”. The answer is more than simply “Connect wire A to termi-nal B....”. It involves an appreciation of the direction in which the experiment is leading and thepoint at which the experiment will be completed. Make the measurements but know why you aremaking those measurements. As an example, think about the measurement of the accelerationdue to gravity, g , using a simple pendulum, something you might have done during your A-levelstudies. The aim is to measure g as precisely as possible within the limitations of the equipmentwith which you are provided. What will those limitations be? (Small angular displacement so thatsinθ = θ , treating the pendulum bob as a point mass, and the string as weightless, ignoring air re-sistance). What will be your objectives if you are to obtain the best possible result? (To measure




the period of oscillation as accurately as possible, taking account of the time available within whichthe experiment will be performed. To measure the length of the pendulum as well and as accu-rately as you can). How will you fulfil those objectives? Is there a point at which improving theprecision of measurement might be considered as counterproductive? What do you choose asthe datum position defining the start (or end) of a swing of the pendulum? Should you time justone oscillation, or five, or fifty or more? How many repeat timings of that number of oscillationsshould you perform? (Remember that you have a finite time in which the experiment has to beperformed.) Is it better to repeat the timing for a given length fifty times or to repeat the timing tentimes for each of five lengths? When you measure the length of the pendulum should you esti-mate the position of the centre of mass of the bob? Or should you measure the length of the pen-dulum to the top of the bob and to the bottom of the bob and assume that the mean of thosemeasurements is the position of the bob centre of mass, i.e., the length of the pendulum. And soon. Details of your decisions and the reasons you made them should be given in the notebook,which provides your record of the experiment.

Then the results of your measurements (including your estimate of the measurement error) are re-corded in your notebook and you will be in a position to extract information from them. Is it best toplot a straight-line graph and measure its gradient? How do you make sure you have drawn thebest straight line? How far may that “best fit” give you an erroneous value of the gradient? Do

the points on your graph (when investigated as single experimental results) all produce the samevalue of g within your estimate of the experimental error?

Finally, you have your measured value of g and confidently state that the value is(9.77±0.08) m s –2. Or more likely (9.5±0.5) m s –2. Then you can compare your result with the ac-cepted value and discuss the discrepancy if they are not in agreement within your estimate of theexperimental error. Is there some systematic error which you have overlooked?

The analysis of the experiment given above is intended to be illustrative, not comprehensive. Your role is active, not passive, and at every stage you should think about what you are doing and whatit means.




3. Your part in the laboratory

3.1 General

The First Year Laboratory is organised to provide you with experience of elementary practical

physics. Each laboratory module consists of 12 weekly three-hour laboratory sessions (11 for PHA182); the first session in semester 1 will follow the maths diagnostic test and will comprise anintroductory talk followed by hands-on experience of the equipment. You should complete one ex-periment in each session, except for the final session, which is for marking of lab books and hand-ing in of reports only. The laboratory experiments you will be asked to perform are designed sothat they can be completed easily within three hours. Your marks will be based on ten experiments(the ten best in the second semester).

Depending on your degree programme, you will take one or two laboratory modules a year in your first two years. The following groups will take the laboratory modules covering the following first-year subjects. Information about the second-year lab is in your programme regulations.

Physics, Engineering Physics, Physics with ManagementPHA181 (Semester 1) Mechanics and Electricity and MagnetismPHA282 (Semester 2) Atomic and Thermal Physics and Waves, Optics and Relativity

Sports Science and PhysicsPHA281 (Semester 2) Mechanics and Electricity and MagnetismPHA182 (Semester 3) Atomic and Thermal Physics and Waves, Optics and Relativity

Quantum Information and ComputationPHA181 (Semester 1) Mechanics , Electricity and Magnetism , Atomic and Thermal Physics andWaves, Optics and Relativity

Physics and Mathematics

PHA285 (Semester 2) Mechanics , Electricity and Magnetism , Atomic and Thermal Physics andWaves, Optics and Relativity

[Physics and Computing have no laboratory until Part B.]

You may sometimes find yourself doing an experiment before the relevant theory is covered inlectures; the laboratory scripts should give you sufficient information to carry out the experiment,but you should ask if you have difficulties. This will give you a useful head start in the appropriatelecture.

A member of the academic staff, who is assisted by one or more graduate demonstrators and atechnician, B K Chavda (Chas), supervises the laboratory. The supervisor and the demonstrators

are involved in the assessment of your work but they, and the technician, are also there to provideyou with support. If you need assistance, e.g., if a piece of apparatus does not work properly or if you don’t understand what is required of you, then ask the staff for their help . If you don’t expressyour difficulties to them there is little possibility of their knowing you are having problems. Don’twaste time waiting until you are approached by one of the staff. Tell them you need help and theywill provide it as soon as possible.

In the laboratory most of you will be paired off to work in groups of two. Each member of a pair must provide an equal input to the work in the laboratory. The two members of a group must pre-sent their work for marking at the same time to one of the demonstrators or to the supervisor. Thelaboratory sheet will indicate who is responsible for marking which experiment.




3.2 The Assessment Process

There are two aspects to your assessment in the laboratory. 70% of the mark for the modulecomes from your laboratory notebook (called lab book in the following) and 30% from twoformal laboratory reports (called report s in the following).

When you join the laboratory you are required to buy from the Department (at the discounted priceof £5) a lab book . During the three hour laboratory class you are expected to enter in this labbook the data you collect from the experiment allocated to you, to plot any graphs, to work out theresults of the experiment, and to present any completed experiment for marking. If possible youshould complete the write-up, by thinking about the meaning of the results and writing down your conclusions, before leaving the lab. You should present your lab book for marking the followingweek if possible; if you need help with the write-up, ask the laboratory supervisor or a demonstrator in good time. The write-up must be completed and presented for marking within two semester weeks of the assignment of the experiment, or on week 12 for experiments carried out in week 11.The deadline for completion of the lab book is the start of the lab session. This tight schedule al-lows you to benefit from feedback on earlier experiments. Each experiment is marked out of 20 sothat the total mark for lab book assessment of the best 10 experiments (all 10 experiments for se-mester 1) will be out of 200. See Section 5 for more information.

The other part of your laboratory assessment is the completion of two reports on experiments youhave already performed and had marked. If there is any part of the experiment you are still unsureabout, ask . You will be told in week 5 (semester 1) or 4 (semester 2) the experiment which will bethe subject of your first report and the report must be handed in during the laboratory session inweek 7 or 6 respectively. The corresponding weeks for your second report will be weeks 10 and12 respectively. The submitted report must be signed in on a class list by the laboratory techni-cian. Each report is marked out of 100, leading to a total report mark out of 200. See Section 6for what is expected in a report, and the appendix for a sample.

Marks are recorded in a card index, which is kept in the laboratory. Please ask the supervisor if you would like to check your marks. You will also be given your mark at the end of the module.We take great care to ensure accuracy; if you believe there is a discrepancy in our transcription of marks or our arithmetic, let us know immediately.

You will need a total mark of at least 40% to pass this (as any other) module. If you get less than30% in a first-year lab module, you will not only fail the year but will not be able to resit inthe Special Assessment Period (summer resits), as you will not have the opportunity tocatch up on missing experiments. You would only be allowed to repeat the lab in the fol-lowing year, and will not be able to proceed to Part B until the year after that. Those whoattend regularly and submit work on time will find it difficult to fail!

3.3 Deadlines

Experiments and reports must be submitted for marking within two semester weeks of their alloca-tion (or one week for the experiment assigned in week 11), and must be completed by the start of the relevant lab session. This will give you useful feedback for the next assignment. Work sub-mitted late will receive a mark of zero , unless an extension has been agreed due to good cause,e.g., illness or bereavement. In such cases, you are advised to discuss the circumstances with thelaboratory supervisor as soon as possible. Given extenuating circumstances, work may be sub-mitted up to a week late without formalityby prior arrangement with the laboratory supervisor . For longer extensions, you must obtain an Impaired Performance form from Mrs Maureen McKenzie inthe Departmental Office (W212). The form, when completed, must be delivered to the Admin 2building. Additional evidence such as medical certificates must be submitted to the DepartmentalOffice. If no such arrangement is made a mark of 0 will be given for that experiment. (See the De-

partmental Coursework Guidelines for further information.) If, due to congestion in the lab, there isno time for the supervisor or demonstrator to mark your lab book on the day, get a demonstrator or

supervisor to sign and date the book to confirm that the write-up is completed, and get it marked atthe next opportunity.




3.4 In Summary

1. In weeks 1 — 11 there is a three-hour laboratory class on one day of each week. Theclass timetabled for week 1 of semester 1 is an introductory talk; that in week 12 is for marking of lab books and handing in of lab reports only.

2. You will attend all sessions and complete one experiment each session. The lab-bookmark is based on ten experiments; in the second semester the best ten will be taken intoaccount.

3. The laboratory is staffed by a supervisor (academic staff), demonstrators (graduates),and a technician. All are there to give help if you require it. Only the supervisor and thedemonstrators are involved in the assessment process.

4. There is a stringent time limit for the presentation of work for assessment. Late submis-sions receive zero marks. See Section 3.5 for the deadlines.

5. The accounts of experiments in lab books must be submitted for marking within two se-mester weeks of the assignment of the experiment, or at the laboratory session in week

12, whichever is sooner. The account must be completed by the start of the lab session.6. Reports must be handed in for marking at or before the lab sessions in weeks 7 and 12 of

the first semester and weeks 6 and 12 of the second semester.

7. Impaired Performance forms must be submitted if there is good reason for a late submis-sion or non-submission of lab books and reports for assessment.

3.5 Deadline summary for semesters 1 and 2 (semester 2 provisional)

SEMESTER 1 SEMESTER 2

Week Starting Experiment Report Starting Experiment Reportdone marked done marked1 4 Oct NO LAB — 7 Feb 1 —2 11 Oct 1 — 14 Feb 2 —3 18 Oct 2 — 21 Feb 3 14 25 Oct 3 1 28 Feb 4 2 1 assigned5 1 Nov 4 2 1 assigned 7 Mar 5 36 8 Nov 5 3 14 Mar 6 4 1 due7 15 Nov 6 4 1 due 18 Apr 7 58 22 Nov 7 5 25 Apr 8 69 29 Nov 8 6 2 May 9 710 6 Dec 9 7 2 assigned 9 May 10 8 2 assigned11 13 Dec 10 8 16 May 11 912 10 Jan — 9, 10 2 due 23 May — 10, 11 2 dueYour deadline is the date of your group’s lab session in the relevant week.

3.6 Next year

The second-year laboratory will be organised in a very similar way. The laboratory supervisor willinform you of any changes. This handbook will therefore be of use in subsequent years.




4. Experimental Procedure

4.1 On arrival

Arrive on time! Bring your lab book, calculator, writing equipment and this handbook. Sign the at-tendance sheet, giving your time of arrival. This is needed both for safety reasons and as a recordof attendance.

Check the allocation board for the experiment which you will perform during that session. Youcollect the lab script (the instruction sheet) for that experiment from the filing cabinet, if you haven’talready picked it up last week. Then you locate the apparatus and set to work; the map next to thefiling cabinet shows you where you will find it.

Experiments come in different shapes and sizes and each one will present you with new chal-lenges. Here are some general points; see also Squires (1985).

4.2 The preliminary experiment

You should, in most cases, carry out a trial experiment. This will help you to

• obtain experience of the way in which you should perform the experiment. You can thendevise the technique which is best for the most efficient completion of the experiment.

• know the apparatus you have set up is working in the way that you want, i.e., to produce

the information you require. If any equipment is not functioning properly you should informa demonstrator or a laboratory technician who will try to sort out your problem.

• decide on the ranges of variables you need to measure. • make a preliminary estimate of the errors which will arise in your measurements. You will

find sometimes (but not always) that the error arising from one of the measurements youmake is “the dominant error”, in comparison with which all the other errors may be ne-glected.

4.3 Repetition of measurements

It is good practice to repeat measurements several times so that

• the probability of your making mistakes in recording numbers and the readings of instru-ments is reduced.

• you can make an estimate of random errors in your observations.

It is often better to repeat measurements for several different values of a parameter. Plotting agraph will tell you not only the value of whatever physical quantity you are trying to measure, butalso serves as a check on the theory you are using and on systematic errors; if you expect astraight line passing through the origin, but your line misses the origin or is curved, ask yourself why.

4.4 Recording information

You should work out results as the experiment proceeds and draw the appropriate graphs as thedata are recorded. If the results are clearly wrong you can repeat the measurements and re-checkeverything.




If you feel you need new or additional equipment to carry out the experiment you should consult ademonstrator or a technician.

It is essential that you record in your lab book the data collected as you make your observations .Calculations performed using the data must also be entered in the notebook. Do not record resultson loose pieces of paper; ignore any instructions to the contrary printed in the lab book.

All this information should be in your lab book. The instruction sheet with which was given to youshould be stapled in there as well; you don’t need to copy out the instructions, but should note if you have deviated from them for any reason. Consult the supervisor if you intend to make signifi-cant changes to the procedure.

4.5 Marking

You will probably have a completed experiment to be marked. In some sessions (see section 3.5)you will be handing in reports as well. Check who will be marking the experiment and arrange,with your partner, to have your books marked at a convenient time during the lab session. As thestart of the lab is usually a busy time for the supervisor and demonstrators, no lab books will be

marked in the first half hour. It is also not a good idea for everyone to present books for markingfive minutes before the end of the session! The marking process should typically take about tenminutes. You will gain useful feedback; be prepared to take notes.

The marking deadline is two semester weeks after the allocation of the experiment (but one weekfor the last experiment; see section 3.5), although that shouldn’t discourage you from getting thelab books marked earlier. You should arrive with the write-up in your lab book complete and readyto be marked; do not waste laboratory time by finishing off the write-up from a fortnight ago . If there is any point you need help with, ask the week before the deadline. Section 5.4 gives an ap-proximate basis for the marking.

4.6 Before you leave

Do as much of the analysis as you can. Make sure that your results make sense and that youhave made all the measurements you need; once the apparatus has been dismantled, it’s too late.Compare your results with any values you find in data books in the lab; you will find it extremelyuseful to buy a copy of the Science Data Book (Tennent, 1971); this will also be provided on your desk in examinations, and it is useful to get to know your way round it first. If you do look up avalue, it is important to make a note of the source. See Section 5 for guidance in what to put inyour lab book.

Get any outstanding experiments marked by the person indicated on the board. If the marker isnot available, have the book signed to indicate that you have completed the write-up and get itmarked at the next opportunity. Find out which experiment you are doing next week and pick upthe lab script so that you can get off to a flying start.

Sign out on the attendance sheet.




5. Laboratory notebooks

Your lab book is a record of the experiment as you proceed; it should contain all your raw data andcalculations, as well as the analysis and discussion of your results. Each experiment should startwith a title and date and the name of your lab partner. Here are some brief notes on graphs, unitsand errors; further details are in sections 7–9 and will be covered in the Information Skills module.(This module is part of the Physics , Engineering Physics , Physics with Management and QuantumInformation and Computation programmes; if you are on another programme in the Departmentyou may find it useful to sit in on the lectures on error analysis, which will take place in the first fewweeks of semester 1.)

5.1 Errors

All experimental quantities, apart from the counting of small numbers of objects, are subject to er-ror. The result of a measurement must therefore indicate the range over which it is likely that thetrue value lies. You should write measured or calculated values in the style “(quantity ± error)units”, e. g., g = (9.5±0.5) m s –2. Remember when you give values with errors that, in general, youshould

• quote the value of the measurement to the position of the most significant figure of the error (or the second most significant figure if the first is 1 or 2).

• quote the error only to its most significant figure (or the second most significant figure if thefirst is 1 or 2).

Details of error estimation and calculation are given in section 7; see also Taylor (1997), Topping(1972) or Barford (1985). For the first couple of experiments, estimate the most important contri-bution to the error and the effect on the result. Once you have covered error analysis and combi-nation of errors in Information Skills (a summary of which is in section 7), you are expected to usethis knowledge in your experimental write-ups.

5.2 Graphs

You should draw graphs on the graph paper in the laboratory notebook, except for logarithmicgraphs, which should be plotted on the graph paper provided on top of the filing cabinet. A graphshould make full use of the page available, but with a sensible scale such as 1, 2, 5, 10, … unitsper centimetre. Axes should be clearly labelled and the graph should be titled. The recommendedformat for axis labels is “quantity/unit”, e.g., “Voltage/V”. The axis label should not contain an error,e.g. not “(Voltage ± 0.1V)/V”; instead, error bars should be plotted for all points where appropriate.When the scale of the graph is such that the errors cannot be plotted, write a statement such as“On this scale error bars are too small to plot”. Be careful that your labels are not ambiguous. For example, “Voltage × 10

–3/V” or “Voltage/× 10

–3V” can mean either that the figures on the axis have

been multiplied by 10 –3 or that they must be multiplied by 10 –3. This will change your numbers by a

factor of a million. You should use “Voltage/mV” to avoid this possibility of confusion. When thereis more than one plot on a graph you should mark their points differently and differentiate betweenthem in the legend. You should avoid writing or entering numerical calculations on the graph. Seesection 8 for further information on graphs.

5.3 Units

You must give the units with all measured and calculated quantities (unless the quantity is dimen-sionless, i. e., a pure number). It’s good practice to write down the units for every intermediate stepin the calculations. Use SI units wherever possible, and always with the final result; it may some-times be more convenient to work in centimetres and grammes and then convert into metres and

kilogrammes in the final line. Table columns and graph axes should be labelled “quantity⁄unit”, e.g. “l ⁄m” for a length measured in metres. Further information and a table of SI units are in section9.




5.4 Marking

You will be marked both on the content of your lab books and on the understanding you show dur-ing the marking process. Lab books are marked out of 20. A mark of 14 or more corresponds to afirst-class performance (if continued consistently throughout the module) and 7 or less a failure.When lab books are marked, approximately equal weight is given to each of the following: 1. Experimental technique and care taken in measurements. 2. Clear recording and presentation of results, including good choice of graphs to draw and accu-

rate, clear plotting of graphs (title and legend, axes labelled with numbers and units, pointsclearly marked, lines drawn smoothly, no extraneous writing on graph, etc.)

3. A careful discussion and analysis of errors, their origins, their magnitudes, and their relative im-

portance. 4. Showing an understanding of the experiment, its background and its implications. Should you

get a nonsensical result (it happens!), check your calculations carefully and comment on the re-sult.

5. Clarity and completeness of the notes in the lab book. You will be recording the experiment in your lab book as you proceed. For this reason, neatnessand intelligent organisation of presentation in the lab book, while desirable, are not essential, pro-vided that your work is legible and it is clear what you have done.

5.5 Plagiarism

The University takes a serious view of plagiarism. For more information see the Departmentalpolicy on plagiarism on p 14 of the Departmental handbook and the University policy in the studenthandbook. Feel free to discuss the experiments with anyone, but do not copy any results fromanyone other than your lab partner. When writing a report, you must not copy from your lab part-ner either. Any information from an outside source must be cited. Even if you just look up valuesin a textbook or data book or web site, for example to compare with your results, it is important tocite your source (although this is more a matter of making checking easier).




6. Reports

6.1 Aim of the report

A vital skill of the physicist (and of most other professions) is the ability to communicate the resultof your work. You must develop the ability to describe, in writing as well as orally, the experiment

you have performed, to explain why it was done, what results you obtained and what they mean, ina form that others can understand. The conventional forms of written description of your workwould be a scientific paper, or an internal company report, or perhaps an article for a publicationsuch as New Scientist . Most of you will learn more about report writing in theInformation Skillsmodule.

Your report will consist of several related sections. After the title page the sections should be Ab-stract , Introduction , Experimental Method , Results and Discussion , Conclusions and References .We are being prescriptive in asking you to use this structure not to restrict your imagination but topromote uniformity of marking and to ensure that all information is present. Your approach tothese sections should be guided by the outlines given below.

The report should preferably be word-processed. An advantage of doing so is the practice gainedin the useful skill of scientific word processing. If you hand-write a report ensure that it is legible.To find if what you have written makes sense and has the meaning you intended, read your workout loud to yourself. You might give it to a friend who hasn’t done the experiment — preferably onefrom a different department — to see if it makes sense. Note the University’s plagiarism policy;you must sign the declaration on the cover sheet that your report is your own work. Anyinformation you get from a book, the Internet, etc must have its source cited, and any help fromfriends must be acknowledged. It is acceptable for you and your lab partner to collaborate in writ-ing up an experiment in lab books, but you must write separate reports. Complete and sign thecover sheet before handing the report in; the member of staff who collects it will sign and return thereceipt.

The format you should use for the presentation of laboratory reports is described below. Its style istypical of a paper or a report. A brief sample report is given in the appendix. Always bear in mind that this is an exercise in communication; write for a reader who is not familiar with the experiment you have done. See Ebel et al (1987) or Farr (1985).

6.2 Format of report

6.2.1 Language and style

Some of the marks are for your use of the English language (spelling, grammar and punctuation).Don’t rely on spell-checkers and grammar-checkers; see for example Trask (1997). The reportshould be written in connected prose, not as a sequence of equations. Equations and numerical

results, although usually on a line by themselves, should be parts of the surrounding text andpunctuated accordingly:

Using Einstein’s (1905) result that the energy of a body of mass m at rest is

E = mc 2, (1)

where c = 3 × 108 m s –1 (Tennent 1971) is the speed of light, we find the energy of the sample to be

E = (1·5 ± 0·3) × 1017 J. (2)

In a report, when describing work that was done, always use the past tense, e.g. “The voltage wasmeasured...” rather than “The voltage is measured...” or “Measure the voltage”. However, whendiscussing the results presented in the report use the present tense, e.g. “These results show...”.It’s better to write your report in a passive form, e.g. use “The voltage was measured...” rather than“We measured the voltage...”. (Objections raised by grammar checkers on the use of the passive




should be ignored.) Never use “I” even if your lab partner didn’t turn up and you did the experimenton your own; “we” is also best used when referring to both author and reader, e. g. “from theseresults we see that…”.

6.2.2 Title Page

Give the title of the experiment, your name, the date on which the experiment was performedand the date on which the report is due.

6.2.3 Abstract

The abstract is a very short (< 150 words) self-contained summary of the report. It is in-tended to be read in complete isolation from the rest of the report. You must make no refer-ence to figures or tables given in the main body of the report, or to the lab script. You shouldstate briefly what was done in the work and the main result(s) and conclusions. (The abstract of a paper in a scientific journal is often more freely available on the Internet and elsewherethan the full text, and allows readers to determine what the results are and whether it is use-

ful to read the paper.) For example,The acceleration due to gravity, g , has been measured by timing the fall of a piece of chalk with ahand-held stopwatch. The result obtained is g = (9.4±0.4) m s-2. This is in reasonable agreementwith the previously reported value1 of 9.8 m s-2.

6.2.4 Introduction

You should introduce the topic of the experiment to the reader, i. e., some background to thesubject will be required in an interesting and informative way. Explain concisely why thephysical phenomenon under examination is important in, e. g., the construction industry,electronic engineering or the study of subatomic particles. You might like to put the experi-ment into historical perspective. This will need some background reading. If you use books,

journals, etc., you must cite these. You should also explain the aims of the experiment.

The introduction should give in outline the theory relevant to the experiment. This theoryshould make clear the origins of the formulae you use, but should not include every step in amathematical derivation of the formulae in question. Don’t give the theory in infinitesimaldetail. This doesn’t mean you should state only the formulae used to analyse the informationyou gather; more is required than that. Equations always need to be in context: you need todefine the quantities involved, and explain the range of validity and the area of application,before or in the same sentence as the equation. Be careful when making forward referencesto the experiment; don’t say “the voltmeter” before you’ve told the reader that a voltmeter isto be used.

6.2.5 Experimental Method

This is where you describe what it was you did when you performed the experiment. Askyourself how the experiment was done, what instruments and apparatus were used, andwhat observations were made. Though the report is written impersonally you must not writeit as a set of instructions appended to a list of the apparatus used. Write “The length of apiece of string was measured”, not “Measure the length of a piece of string”. You shoulddraw carefully labelled diagrams (with the figure number and caption directly below) to helpyour reader to understand what you have written. No marks are given for copying out the labscript; you must write the experimental procedure in your own words, and not necessarily in

the order it appears in the script.




6.2.6 Results and Discussion

You should list the information gathered in your experiment in tables summarising the re-sults, e.g. at most give the average times observed for 50 oscillations of pendulums of vari-ous lengths rather than a listing of all the times measured. It is a waste of time to copy outlarge tables from your lab book and, if you can draw a graph, you don’t need the table aswell. When you use graphs mark the points clearly so that the quality of the data is immedi-ately apparent. Comment on any spurious points and unexpected deviations from the antici-pated results. State how your results have been calculated, and give enough information for the reader to be able to check your results if necessary, but don’t repeat every line of the cal-culations. A brief account of, and the results of, error calculations should also be given. Al-though it is too late to carry out further measurements, by all means try further analysis of your results, for example by means of a spreadsheet. If, when your lab book was marked,you became aware of a mistake in the analysis, you have a chance to regain your lab credby getting it right in the report.

6.2.7 Figures and tables

You should number figures (and graphs and tables) and give them captions. By reading thelegend the reader should be able to understand the figure without having to refer to the maintext. It is best if you can embed the figures, tables, and graphs in the report close to theplace where reference is first made to them. You could also draw the figures on separateleaves and insert them in your report next to the page on which you first refer to them (moreconvenient for you if you can’t easily draw them electronically). It is acceptable, but not soconvenient for the reader, if you gather the figures in numerical order at the end of the report.

6.2.8 Conclusions

You complete the text of the report by tying together the main themes of the work. Show how

the results reported fit within the context of the subject. You may think it appropriate to indi-cate areas for further work or ways in which the experiment could be improved, e.g., how er-rors might be reduced.

6.2.9 References

You will have referred to any text books, lab scripts, data books, journal articles or web sites,etc., in producing your report and you must include them in your list of references. There aremany formats in common use for references and the order of items differs, but they must in-clude author, title, date of publication and publisher for a book, and author, journal title, vol-ume, page and date for a journal article. (If you submit your paper for publication, you must

use whatever format the publisher prefers.) You must cite your references where they areused in the text using either the numerical or the Harvard alphabetical systems:

1. Numerical system: you number each reference and refer to it in the main text using asuperscripted number 1. The references should be numbered in the order in which theyappear in the text.

1 I Newton, Philosophiæ Naturalis Principia Mathematica p 13 (Roy Soc, 1686)

2. Harvard alphabetical system: References are listed alphabetically and are referred to inthe main text by name and date, as “Einstein (1905) showed that…” or “Recent obser-vations (Galileo 1610) have challenged the geocentric theory…”.

Einstein A, (1905) Annalen der Physik 17 639.




6.3 Assessment

Reports will be marked following the guideline marking scheme given below. The questions act asprompts for the person marking the report.

General Presentation 10%

Is the report neat? Are all the appropriate sections present and labelled cor-rectly? Do figures and tables have numbers and legends? Are figures andtables referred to in the text? Are the figures and tables in an appropriate for-mat?

Use of English 10%What is the standard of English grammar? Are tenses used appropriatelythroughout the report? Is the report written in the correct person?

Abstract 10%Does the abstract state briefly, clearly, and comprehensively, what was done?Is the abstract self contained?

Introduction 20%Does the introduction set the scene for the reader who is less well informedthan the writer? Is the wider context of the work described? Are the aims andintentions of the experiment given? Is the theory necessary to understand theexperiment given? Is the level of detail suitable? Are all terms defined?

Experimental Details 15%Is a full yet concise description of the equipment and procedure given? Is ap-propriate use made of diagrams?

Results and Discussion 20%Is appropriate use made of figures and tables to present data? Are errorstreated in an appropriate fashion? Are the results fully discussed? Are the re-sults compared with those that were expected?

Conclusions 10%Is the work summarised? Are numerical results presented again? Is therefurther discussion of the results?

References 5% Are references cited at appropriate places in the text? Are references given inthe text in a recognised style? Are references given in the reference section ina suitable style?

Total 100%



First Year Laboratory Handbook – 18 – RTG 18 September, 2004

7. Measurement Error

7.1 Introduction

Generally speaking, when we make a measurement, we would like to know the true value of thequantity we are measuring. When the measured quantity is a whole number, like the number of people in a room, this ideal can indeed be realised. However, for most kinds of measurements, noexperiment will yield an exact result, no matter how good the experiment, and in such circum-stances the best the experimenter can do is to quantify the uncertainty or “error” in the result. Nor-mally this means presenting the result as a “best estimate” x 0 plus or minus an “error” δ x :

x 0± δ x meaning that the true value probably lies in the range between x 0 – δ x and x 0 + δ x with the mostlikely region being centred on x 0; we define δ x so that the probability for the true value to lie in thisrange is 68%.

In everyday life people rarely quote error estimates on the measurements that they make such as

the outdoor temperature or the speed of their car. This is because all they are doing is reading thedial of a commercially produced instrument which has been specifically designed to measure thequantity of interest to the level of accuracy which is generally required. Furthermore any goodcommercial instrument will be designed so that it is accurate to better than the precision with whichthe scale can be read. So if someone tells you that a commercial thermometer reads 19 °C thenyou can be pretty certain that the temperature is greater than 18.5 °C and less than 19.5 °C.

However as an experimental scientist you will make much more complex measurements than sim-ply reading a dial. Furthermore there is frequently no “required” level of accuracy; your job is tomake a good measurement (including an error estimate) which can be used by other people for whatever purpose they wish. You should therefore never present a measurement without alsosaying something about the likely magnitude of the error.

This document has two aims:1. To help you to give sensible error estimates on your measurements.2. To help you understand how error estimates are used in the interpretation of experimental re-

sults.

7.2 Drawing conclusions from experiments

Before proceeding to the rather involved business of estimating the magnitudes of errors it will beinstructive to take a brief look at how error estimates are used in interpreting experimental results.

After all, the very purpose of making error estimates is so that we can draw conclusions from our experiments.

Suppose a quantity x is measured as12±20

i.e. x probably lies between –8 and 32. We can now compare our measurement with predictionsabout the true value. Suppose we have two theories which predict the values of x given in Table 1.What can we conclude?

_______________________________________________________________

Prediction for x Theory A 0Theory B 100

Table 1Predictions from two theories




Let us take theory A first. Certainly we can say that theory A is consistent with our measurement .Have we proved that theory A is correct? The answer is emphatically NO! It may well be that amore accurate measurement of x would be inconsistent with the prediction from theory A; further-more even if theory A is absolutely correct in its prediction for the value of x it may well be wrong inits predictions for the values of other quantities. Thus we see that no theory is ever proved correct;there may always be another experiment around the corner, which will show it to be false.

Now let us take theory B. We note that the prediction from theory B lies well outside the statedprobable range for x . Have we proved theory B to be incorrect? Well........ not absolutely. Theresult of the experiment was only that the true value of x probably lies in the range between –8 and32 so there is still the possibility that the true value lies outside this range. However, assuming thatthe error estimates are correct the probability that the true value of x lies more than 4 δ x from x 0 isexceedingly small; Table 2 gives a rough guide to the probabilities involved and in our case theprobability is given as 6.3±10 –5 so either theory B is incorrect or the experimenter has been ex-tremely (!) unlucky to make such a bad measurement. So we can conclude that the experimentcasts very serious doubt on the validity of theory B; if a few more experiments came up with similar results we would completely forget about theory B.

∆ P δ x 0.322 δ x 0.0453 δ x 0.00274 δ x 6.3× 10 –5

5 δ x 5.7× 10 –7

Table 2ProbabilityP for the true value to lie greater than a distance ∆ from x 0.

(assuming a Gaussian probability distribution of standard deviation equal to δ x)

Thus for the experimenter trying to test a theory the purpose of estimating errors is in order to ex-

tract a probability from Table 2 and hence make a judgement about whether the experiment rulesout the theory or not. Clearly it is not critical to be extremely accurate in estimating errors and weare never worried about estimating errors to better than 10% accuracy.

7.3 Presenting a measurement It is the accuracy of the error estimate that determines how many decimal places to use in pre-senting a measurement. Thus if your best estimate of the true value is 2.185617 and your esti-mate of the error is 0.015 then you should quote your result as

2.186±0.015The extra digits that have been thrown away here really do not contain any useful information. Inpractice you will neither throw away much useful information nor keep much useless information if you blindly follow the following rule for how many significant digits to use in quoting the value of theerror:

no. of significant digits =

two if first digit is 1 or 2

one otherwise

The following are examples of following this rule: 5.93±0.12, 5.93±0.29, 5.9±0.3, 5.9±0.5, 5.9±0.9.

If powers of 10 are involved it is clearer and simpler to write, for example(2.186 ± 0.015)× 10 –12

rather than2.186× 10 –12 ± 0.015× 10 –12.

7.4 Random and Systematic Errors




You know very well that if you repeat an experiment you do not normally get exactly the same re-sult as you did before: a sequence of many measurements fluctuates randomly about some centralvalue. This kind of error is called random error (or statistical error ). There is a multitude of pos-sible causes (e.g. vibration, electrical noise, observer variability etc.).

Imagine repeating a measurement a large number of times. Having made the measurements youcould then draw up a table of how many measurements are in different intervals; it might look likeTable 3.

x interval Number of measurements

3.75 – 4.25 04.25 – 4.75 14.75 – 5.25 05.25 – 5.75 55.75 – 6.25 46.25 – 6.75 66.75 – 7.25 27.25 – 7.75 27.75 – 8.25 08.25 – 8.75 08.75 – 9.25 0

Table 3Some sample data

x

0

0.1

0.2

0.3

0.4

0.5

0.6

4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9

Figure 1Histogram of the data in Table 1. The quantity plotted is the number of entries divided by the bin

width w and the total number of measurements N .




x

0

0.1

0.2

0.3

0.4

0.5

3 5 7 9

Figure 2 A picture of what Figure 1 might look like if much finer bins were used and an extremely large

number of measurements were made.

You could then plot these measurements in a “histogram” as in Fig 2. Imagine making an infinitenumber of measurements (easier to imagine than to do!) and histogramming them in extremely finebins. The shape of the histogram would be a smooth curve (perhaps the curve shown in Fig. 3)and the mean of the measurements would be well defined, i.e. another infinite set of measure-ments would produce exactly the same curve and exactly the same mean value. The curve shownin Fig 2 is called a Gaussian or normal distribution; most repeated experimental measurementsyield a curve of approximately this shape. Indeed the numbers given in Table 2 assume a curve of exactly this shape.

The mean of an infinite number of measurements is not in general equal to the true value. Anysource of error which prevents the mean approaching the true value is called a systematic error .

A frequent cause of systematic error is imperfect calibration of the measuring equipment itself.

Random Errors

For an experiment in which there are random errors but no systematic errors an experimenter whopresents a result as x 0±δ x is saying “if I had had the time or inclination to repeat the whole experi-ment over and over again, I estimate that 68% of the measurements of x 0 would have been withinδ x of the true value”.

The nice thing about random errors is that the magnitude of the error is easily estimated and easilyreduced. We can see this as follows.

Suppose we take N readings of x , x 1, x 2, x 3, ............ x N .Let x i be the i th reading. We can calculate the mean

x =

1

N x

i

i = 1

N

∑ , (1)

where

xi

i = 1

N

∑ = x1

+ x2

+ x3

+ ........ + x N ,

and we can also calculate the standard deviation σ , where

σ 2

=

1 N

( x i − x )2

i=

1

N

∑ . (2)




Imagine creating a histogram as before: if N is large then the histogram will follow a smooth curveas before. Using our histogram we can find the value of δ x using our definition that 68% of thedata lies between x 0 –δ x and x 0+δ x . If our histogram has a Gaussian shape then we would find that

δ x ≈ σ .Since σ is easy to calculate we use this method of calculating the error δ x . If you don’t have such aspecial facility on your calculator for calculatingσ you might find the following formula easier to usethan equation 2:

σ 2

=

1

N x

i

2

i = 1

N

∑

− x

2 . (3)

Many people's definition of error is directly in terms of standard deviation rather than our definition,which is in terms of probability; in practice it, makes little difference.

Hence the value of δ x that we have calculated is the error on each one of our N measurements of x , i.e., we have N measurements and we know the error on each of them:

x 1± σ .

x 2± σ .

x 3± σ ...

x N

± σ .We have already indicated that the mean x provides a better measurement of the true value thanany single measurement x

i and a good estimate of the error in the mean is σ / N , i.e. the error in

the mean is N times smaller than the error in a single measurement. Note that as N → ∞ theerror in the mean tends to zero as anticipated earlier. Thus, neglecting systematic errors, weshould present our best measurement of x as

x ±σ

N . (4)

If there is also a systematic error s affecting all the N measurements equally, then following therules given below (equation 6) the result should be presented as

x ± s2

+σ

2

N . (5)

Unless you are confident that random errors are unimportant you should always repeat a meas-urement several times (at least 5) so as to estimate the size of the random error and improve theaccuracy of your measurement. If your experiment is dominated by random error you should makeas many measurements as you can stand (10 or 20).

Systematic Errors

We have defined a systematic error as any source of error that causes the mean of a large number of measurements to fail to converge to the true value. Unlike random errors there is frequently noeasy way of estimating systematic errors at all accurately; furthermore the only way to reduce themis to improve the experiment in some way.

Perhaps the most common cause of systematic error is the error in the calibration of a measuringdevice (voltmeter, thermometer etc.). The simplest example of this kind of error is a zero offset(e.g., bathroom scales suffer from this so much that there is usually a knob you can adjust to mini-mise it). The manual of any measuring device will specify (in the “Specifications” section) the ac-curacy that can be expected. In the absence of a manual it is a fairly safe bet that the manufac-turer has designed the calibration error to be less than the error in reading the scale, and so thecalibration error can probably be ignored as negligible. The scale reading error normally contrib-utes to the random error so if you have estimated random errors by taking repeat readings thenyou have probably already taken the scale reading error into account.




Because systematic errors are hard to estimate it is normal practice to err on the side of overesti-mation rather than underestimation of the error. By overestimating errors one is trying to ensurethat one’s measurements are not used to draw false conclusions. However one can ruin a goodexperiment by thoughtlessly and unnecessarily inflating error estimates.

7.5 Adding Errors

There may be more than one source of uncertainty contributing to the error in a measurement. For example there may be the random error (determined from repeated measurements) and one or more sources of systematic error.

Suppose that in a measurement of a quantity x there are three kinds of error estimated as δ x a , δ x

b

and δ x c. If these sources of error are completely independent (i.e. uncorrelated) of one another

then they can be combined into a single error estimate δ x according to the following formula:δ x 2

= δ xa

2+ δ x

b

2+ δ x

c

2 . (6)When we add squares like this we say that we are adding in quadrature . Beware that this formuladoes not apply to correlated errors so be on the look out for correlations and try to avoid them if you can — ask a demonstrator if you suspect that you are adding errors that are not independent

of one another.Note that if there are just two sources of error δ x

aand δ x

b then the rule for finding the combinederror δ x is exactly the same as the rule for finding the hypotenuse of the triangle in Fig 3

_________________________________________________________________

δ x

b

a

δ

δ x

x

Figure 3The triangle rule for adding uncorrelated errors

_________________________________________________________________ The fact that errors add in this way leads to a great simplification in the combining of errors. Sup-pose for example that δ x

a= 1 , δ x

b= 0.2 and δ x

c= 0.2 . Combining these errors according to

equation 6 gives us δ x = 1.04 . We see that the combined error is very close in magnitude to thelargest single contribution δ x

a , and practically speaking it would not have made any difference hadwe neglected δ x

b and δ x c

— since we are not interested in estimating errors to better than 10%accuracy. In this case δ x

ais called the dominant error as it makes by far the dominant contribu-

tion in determining the combined error δ x .

From this we draw some important conclusions:1. In any experiment we should always be concerned to identify the largest sources of error; it is

quite likely that there is just one dominant source of error.2. Large sources of error need to be accurately estimated whereas small sources of error onlyneed to be very roughly estimated.

3. In trying to improve an experiment one should focus on reducing the dominant error.4. While the magnitude of every possible contribution to the combined error needs to be esti-

mated it may well be sufficient to assume that the combined error is equal to the dominant er-ror.

7.6 Error in a function of a measured quantity

Suppose a quantity x is measured to be x 0±δ x and suppose f is a function of x , i.e. f =f ( x ). Clearly if

there is uncertainty about the true value of x there is also uncertainty about the true value of f . Wewould like to calculate a best estimate of f together with an error estimate δ f . Suppose the graph of f(x) is as in Fig 3




x

f

xo- δ x xo xo+ δ x

f(xo+ δ x)

f(xo)

f(xo−δ x)

Figure 3The error in f due to an error in x

_________________________________________________________________

Our definition of the error δ x is that there is a 68% probability that the true value of x lies in therange between x 0 –δ x and x 0+δ x . There is therefore also a 68% probability that f lies in the rangebetween f ( x 0 –δ x ) and f ( x 0+δ x ). The best estimate of f is f ( x o). There is a slight problem in assign-ing an error to f because the curvature in f(x) has caused the difference f ( x 0+δ x ) – f ( x 0) to be sig-nificantly less than the difference f ( x 0) – f ( x 0 –δ x ). However normally errors are sufficiently smallthat the effect of any such curvature is negligible so we can make the approximation that

δ f ≈ f ( x0+ δ x) − f ( x0 ) (7)

or equivalentlyδ f ≈ f ( x0 ) − f ( x0

− δ x) (8)or, using calculus,

δ f ≈ δ xdf dx

x = x 0

. (9)

The modulus signs are necessary because it is meaningless for the error to be negative.

Thus we present our measurement of f asf ( x 0) ±δ f ,

where δ f is calculated from one of the above three formulae.

ExampleIf x is measured as 3.0±0.3 what can we say about y =2 x 2?

Let y 0 be our best estimate of the true value of y and let δ y be the error in y. From what we havesaid above we obtain y 0=18. Using the above three formulae our three estimates δ y of come outas 2(3.3 2 –3.02)=3.8, 2(3.02 –2.72)=3.4 and 0.3 × 4× 3.0=3.6. Thus we present our measurement of y as

18±4.




7.7 Error in a function of more than one measured quantity

Let f be a function of x , y and z : f =f ( x,y,z ). Suppose we have measured x , y and z as respectively x 0

± δ x , y0± δ y and z 0

± δ z , and we want to calculate the corresponding measurement of f : f 0 ± δ f .

Not surprisingly, f x0 , y0 , z0( ) is our best estimate of the true value of f . Calculating the error in f presents slightly more of a problem. The errors in the measurements of x , y and z each producean associated error in f . Let δ f x , δ f y and δ f z be the error in f due to the errors in x , y and z respec-tively. We can extend our analysis in the previous section to derive

δ f x ≈ f ( x0+ δ x, y0 , z0 ) − f ( x0 , y0 , z0 ) ≈ δ x

∂ f ∂ x

x = x0 , y = y0 , z= z0

(10)

δ f y ≈ f ( x0 , y0+ δ y, z0 ) − f ( x0 , y0 , z0 ) ≈ δ y

∂ f ∂ y

x = x0 , y = y0 , z= z0

(11)

δ f z ≈ f ( x0 , y0 , z0+

δ z) − f ( x0 , y0 , z0 ) ≈ δ z

∂ f ∂ z

x= x0 , y = y0 , z = z 0(12)

where∂ f ∂ x

means partial differentiation of f with respect to x , i.e. when differentiating f with respect

to x we treat y and z as constants. If these three sources of error in f are independent we can addthem in quadrature according to equation 6, obtaining

δ δ δ δ f f f f x y z 2 2 2 2

= + + . (13)

ExampleLet x , y and z be measured as 3.0±0.3, 2.00±0.10 and 40±4 respectively. Let f = 5 z( x − y) . Whatis the best estimate of f and its error?The best estimate is

f 0 = f ( x0 , y0 , z0 ) = 5 × 40 × (3 − 2) = 200 .Now calculate δ f x , δ f y and δ f z separately:

∂ f ∂ x

= 5 z so δ f x = 0.3 × (5 × 40) = 60 ,

∂ f ∂ y

= − 5 z so δ f y = 0.1 × (5 × 40) = 20 ,

∂ f ∂ z

= 5( x − y) so δ f z = 4 × (5) = 20 .

Thusδ f

2= 60

2+ 20

2+ 20

2= 4400

and henceδ f = 66 .

We therefore say that we have measured f as being 200±70 and note that the dominant error is theerror in x ; the effect of including the other two errors was to increase the error in f from 60 to 66.

7.8 Labour Saving Formulae

The task, described above, of calculating the error in a function of independent measured quanti-ties is a bit laborious so most experimenters find it a good idea to remember the formulae given in




Table 4 which apply in the most commonly occurring cases. There is nothing new in these formu-lae; you can derive them directly using the methods given in the previous two sections.

Function f x y( , ) Expression for error δ f

f=ax+b δ f = aδ x

f=x+y or f=x-y δ f 2

= δ x2

+ δ y2

f=axy or f=ax/y δ f f 0

2

=δ x x0

2

+δ y y0

2

f = ax n δ f f 0

= nδ x x0

Table 4Formulae for evaluating the error δ f in various common functions f ( x, y) .

x and y are assumed to have been measured respectively as x 0± δ x and

y0± δ y . a, b and n are taken to be known constants and f f x yo o o

= ( , ) . _________________________________________________________________

Note the significance of fractional errors, e.g. δ x/x, in calculating the error in a product, quotient or power.

ExampleLet us repeat the example calculation given in the previous section but this time use the formulaein Table 4.

Our function is f = 5z ( x–y ). Let s=x–y so that f=5zs . Using the second formula in Table 4 we cal-culate the error δ s in s to beδ s = 0.3 2

+ 0.1 2= 0.316 (14)

Since f=5zs we can now use the third formula in Table 4 to calculate the error δ f in f :δ f

200

2

=0.316

1

2

+4

40

2

(15)

giving δ f = 66 as before.

Using the previous method it was obvious that the dominant source of error in f was the error in x .To see that in this method we have to start with the last line and work backwards. In equation 15we see that the error in s makes the dominant contribution to the error in f, and in equation 14 we

see that the error in x makes the dominant contribution to the error in s . Thus we reach the sameconclusion as before.



First Year Laboratory Handbook – 27 – DEGW 18 September, 2004

8. Graphs

8.1 Types of graph

It is often very helpful to present results in graphical form. A graph can help you todetect and eliminate experimental errors, and a graph can be scanned ‘at a glance’.

Most frequently results will be compared with some theoretical formula. It is a greatadvantage to arrange the formula so that if it is obeyed the experimental points willlie on a straight line. For example, if the measurements are of y versus x and thetheoretical formula is y = Ax 2, then a graph of y 1/2 versus x is a straight line through

the origin, since y 1/2 = A1/2 x . Equally, y versus x 2 is a straight line and so is ln y versus ln x . This example illustrates two main points: there is not a unique rear-rangement to get a straight line, and there is no set of rules to find such a rear-rangement. The following examples may help.

It can be seen from these examples that both the slope and the intercept can beused to measure unknown quantities. Note too that the choice of linear form de-pends to some extent on which quantities are known and which are unknown. Thusin the example y = A / x + x o( ) , if x 0 is unknown you have to plot 1/y against x , asshown above. But if x 0 is known, you can also plot y against 1/( x + x 0 ). Sometimesnone of this is possible and you have to use a spreadsheet.




Relation Linear Form Graph

y = Ax + C

y = Ax + C

y = Axn

y1/ n

= A1/ n x

(n known)

y = Axn log y = log A + n log x(n unknown)

y = A( x + x0 )n y1/ n

= A1/ n ( x + x0 )(n known, x o unknown)

y = A

x + xo

1

y=

1

A x + x0( )

( x 0 unknown)

y = y 0 10− x / x 0

log y = log y 0 − x

x 0

Fig. 1 Linearisation of graphs by alge-braic manipulation.




8.2 Logarithmic Graph Paper

In the last of the above examples, log 10 y is to be plotted against x . This happens sofrequently that special logarithmic graph paper is available. On log-linear graph pa-per, the vertical scale is proportional to log 10 y , and the horizontal scale is linear.More precisely, the distance between two points y 1 and y 2 on the vertical axis is pro-portional to log10 (y 2/y 1). Note that there is no origin, since log 100 = - ∞ . The verticalaxis is marked in decades , and the lines labelled 1, 10, 100, 1000 ... are equallyspaced, since log 10 (10/1) = log10 (100/10) and so on. The lines 1, 2, 3, 4, … are not equally spaced. Log-linear graph paper comes as 1-cycle, 2-cycle, 3-cycle and soon, the number indicating the number of decades on the vertical axis. It is importantto choose the right paper for your results so as to spread the points over most of thevertical axis. Writingy max/y min as the maximum and minimum readings, then, if y max/y minis between 10 and 100, 2-cycle or at most 3-cycle paper should be used. On theother hand, with y max/y min of order 105, 6-cycle paper will be needed.

To obtain the gradient of your graph (after you’ve plotted it to obtain a good straightline) two points (y 1, x 1) and (y 2, x 2) will need to be selected. As logarithm y is propor-tional to x , the gradient will be

ln y2 − ln y1 x 2 − x1

=

ln( y 2 / y1)

x 2 − x1

so that either ln y 2 and ln y 1 or ln (y 2 ⁄ y 1) will have to be calculated assuming that youwish to work to base e , or log10 y 2 and log10 y 1 or log10 (y 1/y 1) if you wish to work to base 10.

Log-log graph paper is also available, on which both axes are logarithmic. This ismost often used to test for a power law dependence. Thus if y = Ax n, with n un-known, then

log10 y = log10 A + n log10 x

A graph of log10 y against log 10 x (i.e., y versus x on log-log paper) is obviously astraight line with slope n .

Since log-log graph paper has been scaled logarithmically identically in the x and y directions, the gradient can be obtained directly from the gradient on the graph pa-per. There is no need to calculate actual logarithms. If you are unsure, read off twopoints from the straight line you have plotted, put them into the theoretical equation

and solve the two resulting simultaneous equations.

8.3 Axes

Graphs (apart from logarithmic graphs) should normally be drawn on the graph paper in the laboratory notebook. A graph should make full use of the page available.Graphs should be titled. Axes should be clearly labelled. The format for axis labels is“phenomenon/units”, e.g., “Voltage/V”. The axis label should not contain the error.The use of ambiguous labelling should be avoided. For example, “Voltage x 10 -3/V”or “Voltage/ x 10-3V” can mean that the figures on the axis have been or have to bemultiplied by 10-3. This is a difference of 106 times. It is far less ambiguous to write“Voltage/10

-3V” or better still “Voltage/mV”. When there is more than one plot on a

graph the points should be clearly marked with different point symbols. Always markpoints clearly and use error bars when necessary. When the scale of the graph is




such that the errors cannot be plotted, the statement “Error bars too small to plot onthis scale” should be written in the caption.

8.4 Error Bars

In general, the estimated error of the reading should be indicated on the graph. Thisis often done by using an error bar . If the reading is say (2.10 ± 0.10) mA, this isindicated with a short line from 2.0 mA to 2.2 mA. Note that in general the meas-urements of both variables on the graph are subject to error, so that each point willhave both a horizontal and a vertical error bar. In a particular case, of course, it mayhappen that one or other variable is measured with an estimated error that is negligi-ble on the scale of the graph, and in that case there is no error bar in that direction.

Fig. 2 A graph showing error bars for both axes

8.5 Errors in Gradients

The slope and intercept of a straight line graph will often give useful informationabout quantities of interest. When a graph is expected to be linear and the pointsdrawn on it show some scatter there is no longer a unique straight line through them.Many lines could be drawn, all fitting the data more or less equally well. Each of these lines has its own slope and intercept, so that probable errors are associatedboth with the slope and the intercept. You must make estimates of these probableerrors.

Graphical method

Simply draw the ‘best line’ and two ‘extreme lines’ such that the two of them encloseabout 70% of the data points. The difference in slope (or intercept) between the




‘best line’ and the ‘extreme line’ in either direction gives an estimate of the probableerror. An example is shown below.

Fig. 3 The “best” line and two “extreme” lines to fit a set of points.

Sometimes the graph should go through the origin. In these cases, consider whether the zero is just another point subject to error on the graph, or whether it is a point of particularly high accuracy, through which the ‘best line’ and the ‘extreme lines’ shouldpass.

Least squares fittingThis is a means of finding the best fit of some model — for example, a straight line y = mx + c to data points ( x , y ), where m and c are unknown parameters to be deter-mined from your measurements. The formulae used in this straight line fitting andtheir derivation can be found in, for example, Taylor (1997) and will also be dis-cussed in Information Skills . You can fit the straight line plots obtained from your ex-periments using the RM PC in the first year lab: double-click the icon Microsoft ExcelLeast Squares and simply type in your data in two columns in the resulting spread-sheet. It is essential to draw the graph as well — preferably by hand as well as bythe spreadsheet. This will show whether the data look as if a straight line would beappropriate, and will also show up typing errors. You can compare your hand-drawneffort with the machine calculation and so improve your line-drawing technique (andcheck your ability to type in lists of numbers accurately). To gain experience in fit-ting lines by hand, you will not be allowed to use this package until week 7 of your first laboratory semester.




9. Units

9.1 Use of units

It is very important to be consistent in the use of units. All experimental results mustbe quoted with correct units. These must be SI units as listed below, or prefixedunits such as mm; a few quantities may be dimensionless (pure numbers, with nounits). You can’t go wrong if you stick to the following principles:

• Equations must be dimensionally balanced. By the dimensions of a physicalquantity we mean the combination of length, time, mass, etc. making up the quan-tity. For example, in the equation a +b=c the quantities a , b and c must have thesame dimensions and therefore be measured in the same units. If one is a lengthand another is an area you’ve gone wrong somewhere.

• An SI unit for a physical quantity is a fixed reference value with the same dimen-

sions. For example, the kilogramme is the mass of a block of platinum in Paris.When we write m = 5 kg we mean that the mass m is five times the mass of thatblock.

• The solidus (or slash) ⁄ always means “divided by”. For example, the equationabove can be written m ⁄ kg = 5. The object m ⁄ kg (used principally in headings of tables and axis labels of graphs) can be thought of as “mass divided by referencemass”. This is the recommended practice; however, you may often see headingsof the form m(kg), which we do not recommend.

Most people leave them out, but it can help to keep the units in all intermediate stepsin the calculation to check for consistency. For example, if we have an equationg = 2h ⁄ t 2, we might write g = 2 × (4·9 cm) ⁄ (100 ms)2 = 2 × 4·9 × 10 –2 m ⁄ (10 –1s)2 =9·8 m s –2. This way you can do algebra with units: for example, cm ⁄ m = 10 –2. If thequantity has its own unit, such as energy, where the unit is the joule (J), use that inyour final result; if you do not know the name of the SI unit for your result, leave it asa combination (e.g. kg m2 s –2). Never write statements of the form “the mass is mkg”, as m already has dimensions of mass; similarly, do not write “the mass is 5”.Rather write “the mass is m” or “the mass is 5 kg”.

Note for those using word processors: it is conventional to use plain text for unitssuch as m for metres, for standard functions such as sin, exp, log, and for chemicalelements, but to use italic for other scalar quantities such as m for mass, π , etc.Vectors, such as v for velocity, should be bold. All this helps readability; for exampleI might write a formulaI = ma 2, where I might want to refer to a symbol I and a sym-bol a in a sentence. Most equation editors will make the appropriate guesses anditalicise symbols automatically. You should also keep a small space between differ-ent units, for example to distinguish ms –1 (inverse milliseconds) from m s –1 (metresper second). Refer to any textbook for standard style.

In stating experimental results as an equation, you should write the error in the form

x =(1·5±0·3) m .

In a table (such as table 1 below), use one of the two following forms depending onwhether the error has the same value for each data point. Errors should be shownas error bars on a graph (as in figure 1 below) but not stated in the axes. (How might

these data be better plotted?) All scale factors should be combined with the units toavoid confusion; thus you could label the vertical axis as x ⁄ cm or as x ⁄ (10 –2m), butnot as x × 10 –2 ⁄ m, which would mean something different.




t ⁄ ms ± 5 x ⁄ cm10 0.06±0.0320 0.13±0.0430 0.5±0.0640 0.6±0.150 1.4±0.160 1.7±0.170 2.5±0.280 3±0.290 4.1±0.2

100 4.6±0.2Table 1. Height against time of fall for a chalk.

0

1

2

3

4

5

x /cm

0 2 5 5 0 7 5 100 125

t /ms

Figure 1. Plot of height against time of fall. The curve is a guide to the eye.

The SI unit system (“International System of Units”) is the accepted scientific unitsystem. However, it may not always be most convenient to use SI units. The basicphysical quantities are expressed as follows. Note that the name of the unit is inlower case, while the symbol starts with an upper-case letter if it is named after aperson. Also note that the unit of thermodynamic temperature is the kelvin, K, andnot the degree kelvin, °K.

9.2 Basic and supplementary SI units

Physical Quantity Name of SI Unit Symbol for SI unitLength metre mMass kilogram(me) kgTime second s

Electric Current ampere ATemperature kelvin K

Luminous Intensity candela cd Amount of Substance mole mol

The SI also includes two “supplementary” units:




Supplementary PhysicalQuantity

Name of SupplementarySI Unit

Symbol for Supplemen-tary SI Unit

Plane angle radian radSolid angle steradian sr

9.3 Derived units

Units for other quantities are derived from the basic units, and have the names givenbelow.

Derived Name of Symbol SymbolPhysical derived for in SI unitsQuantity unit derived

unit ___________________________________________________________________

energy joule J kg m 2 s-2

force newton N kg ms-2 = J m-1

power watt W kg m2 s-3 = J s -1

electric charge coulomb C A s

electric potential volt V kg m2 s-3 A-1 = J A-1 s -1difference

electric resistance ohm Ω kg m2 s-3 A-2 = V A-1

electric capacitance farad F A 2s4kg-1m-2 = A s V-1

magnetic flux weber Wb kg m2 s-2 A-1 = V s

inductance henry H kg m 2s-2 A-2 = V s A-1

magnetic induction tesla T kg s -2 A-1 = V s m-2

luminous flux lumen 1m cd sr

illuminance lux 1x cd sr m-2 = 1m m-2

frequency hertz Hz s -1

customary degree oC θ c/οC = T /K – 273.15temperature, θ c Celsius

pressure pascal Pa Pa = N m -2

Certain units have proved so popular among scientists that, although there is no fur-ther need for such units, they are nevertheless being retained. Notice that these arenow defined in terms of SI units although originally they were not.

Physical quantity Name of unit Symbol Definitionfor unit of unit ___________________________________________________________________




length parsec pc approx 30.87 × 1015 mångström Å 10 –10 m

area barn b 10 -28 m2hectare ha 10 4 m2

volume litre l or L 10-3 m3

pressure bar bar 10 5 N m-2

mass tonne t 10 3 kg = 1 Mg

kinematic viscosity stokes St 10 -4 m2 s-1

dynamic viscosity poise P 10 -1 kg m-1 s-1

magnetic induction gauss G 10 -4T

radioactivity curie Ci 37× 109 s-1energy electron-volt eV approx 1.6021 × 10-19J

Prefixes

Fraction Prefix Symbol Multiple Prefix Symbol(< 1) (>1)

___________________________________________________________________ 10-1 deci d 10 deca da

10-2 centi c 102 hecto h

10-3 milli m 103 kilo k

10-6 micro µ 106 mega M

10-9 nano n 10 9 giga G

10-12 pico p 1012 tera T

10-15 femto f 1015 peta P

10-18 atto a 1018 exa E10-21 zepto z 10 21 zetta Z

10-24 yocto y 1024 yotta Y

So, for example, the quantity 230 Pa (about 2.3 thousandths of an atmosphere)could also be written as 2.3 mbar, 2.3 hPa, 2.3 × 102 N m-2 or 2.3× 102 Pa.

10. Safety



First Year Laboratory Handbook – 36 – KUN, 18 September, 2004

What YOU should know

You need to develop the habit of asking yourself whether an operation is safe andensuring that you have adequate knowledge and training about safe working prac-tice. Tidiness and thinking ahead are important aspects of safe working practice. For your work in the undergraduate teaching laboratories the following rules apply butyou must use your common sense and ask if you are in any doubt about safety.

10.1 Fire

• The fire alarm is a continuous loud bell. If you hear the alarm you should im-mediately stop what you are doing, leave the building and go to the assemblypoint in front of the main entrance to the Department.

• Fire alarm buttons, fire extinguishers and evacuation procedure notices are lo-cated in all stairwells, as well as at other points in the Department. In the eventof a fire you should raise the alarm and alert a member of staff who will call theemergency services (phone 888).

10.2 General Laboratory Safety

• Students are not permitted to work in the laboratories outside the specifiedtimes of laboratory classes unless they have been given specific permission todo so by a member of academic staff who will then supervise the work.

• Smoking and the consumption of food and drink are not permitted in the labo-ratories.

• Apparatus should be disconnected at the end of a session unless you haveasked for and been given permission by the laboratory supervisor or technicianto leave it assembled.

• All accidents or breakages, however small, should be reported immediately tothe laboratory supervisor or technician. Any dangerous incident or anythingwhich is suspected to be in an unsafe condition must be reported. Cracked or chipped glassware should not be used.

• Coats, jackets and scarves must not be brought into the laboratory but shouldbe left outside on the coat rack provided. Bags should not be allowed to ob-struct gangways.

• Solvents (propanol, acetone etc.) should only be used in a well-ventilated envi-ronment. They are highly flammable and should be kept away from sources of heat.

• Students should be aware of the dangers of loose clothing or long hair whenworking with machinery or chemicals. Strong shoes protect the feet from fallingobjects much better than sandals or canvas trainers.



First Year Laboratory Handbook – 37 – KUN, 18 September, 2004

10.3 Electricity

• Electricity is potentially lethal. Under normal circumstances, any voltage over 55V is to be treated as hazardous unless it is incapable of delivering a currentin excess of 1mA.

• Students are not permitted to work with exposed mains voltages or to performany kind of maintenance work on mains circuitry (including plugs). Equipmentrequiring repair should be brought to the attention of the laboratory technician.

• Before connecting any circuit, a check should be made that all instruments andapparatus are of a suitable rating for the experiment to be performed. Checkalso that all wires are of a suitable current capacity. All supply switches shouldbe in the off position whilst connections are made (during disconnection aswell). All power supplies should be switched off before changing any compo-nents.

• Before switching on check that all connections are correct (if in doubt, ask).Place apparatus so that short circuits cannot occur. When connecting twopieces of mains equipment that both possess earthed terminals it is importantto be aware that these terminals are indirectly connected together; so in gen-eral, when choosing the polarity of your connections you should ensure thatyou connect earth to earth and live to live otherwise the whole apparatus will beearthed and large currents may flow. Also check that any meters you are usingare on an appropriate range setting.

10.4 Lasers

• Lasers can damage your eyes. You must never look into a laser beam or

point laser beams at people. On no account should you stare into a laser beam . The light intensities associated with lasers can readily saturate the de-tection ability of your retina and it is easy to assume that they are not as brightor harmful as they actually are.



First Year Laboratory Handbook – 38 – JHS, 18 September, 2004

References and further reading

N C Barford, Experimental Measurements, Precision, Error, and Truth, 2nd Ed, (Wiley, 1985).

H F Ebel, C Bliefert and W E Russey, The Art of Scientific Writing , (VCH, 1987).

A D Farr, Scientific Writing for Beginners , (Blackwell, 1985).

L Kirkup,Data Analysis with Excel (Cambridge University Press, 2002).

G L Squires, Practical Physics , 3rd Ed (Cambridge University Press, 1985).

J R Taylor, An Introduction to Error Analysis, the Study of Uncertainties in Physical Meas-urements (University Science Books, 1997).

R M Tennent, Science Data Book , (Oliver and Boyd, 1971). ESSENTIAL PURCHASE

J Topping, Errors of Observation and their Treatment , 4th Ed (Chapman and Hall, 1972).

R L Trask, The Penguin Guide to Punctuation , (Penguin, 1997).

The Information Skills module PHA190 includes data and error analysis and presen-tation and report writing. It is part of the Physics , Engineering Physics , Physics withManagement and Quantum Information and Computation programmes; the materialis available on the Learn server.



Appendix: Sample Report

Department of Physics - LABORATORY HANDBOOK- 2004-2005

Documents

Transcript of Department of Physics - LABORATORY HANDBOOK- 2004-2005