SCHOOL OF PHYSICS AND ASTRONOMY

FIRST YEAR LABORATORY PX 1223

Experimental Physics II

Academic Year 2012 – 2013

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Welcome to part 2 of the 1st year laboratory, PX1223. This manual contains the experiment notes needed for this semester; it is expected that you may need to refer to the PX1123 for guidance on use of some equipment and advice with respect to report writing etc.. If you cannot find the information that you are looking for, please ask any member of the teaching team - your Head of Class or the demonstrators. Both manuals are available on Learning Central.

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CONTENTS: I: Logistics of PX1223

Introduction 3

Assessment 4

Refreshments 4 Safety: Risk Assessment and Code of Practice 5 Code of Practice 6 II: Experiments

Timetable and list of experiments 7 Check list for experiments 8

Laboratory notes for experiments 9

III: Background notes

1. Apparatus notes. 60 2. Using Excel 69

3. DIARY and LONG REPORT checklist 71

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I: Logistics of PX1223 INTRODUCTION There are 11 laboratory sessions in the Spring Semester They are designed to extend your skill in, and understanding of, the techniques of scientific measurement and to provide practical experience, where possible, of physics material of the lecture courses and beyond. In addition you will gain more experience in communicating your findings through scientific writing. The majority of the work you will do in the laboratory will be experimental, and will be performed individually. All observations made during an experiment should be entered in your laboratory diary. Each week you will be allocated an experiment and you will normally be expected to complete this, performing appropriate calculations, drawing graphs etc., by 16:00hrs (4pm) on the day following that on which you did the experiment. It is essential that you put aside about ½ hour before you come to the practical class in order to read through the experimental notes associated with the practical that you will be undertaking. This will enable you to gain familiarity with what is expected of you, time to plan your experiment (which will save you time on the day) and very importantly a chance to think about the safety considerations that are required for your experimental work. You will be required to write up one experiment in the form of a formal report, which will be allocated towards the end of the semester. Formal reports should NOT be written in your lab. diary but electronically generated on sheets of paper that are either bound or stapled. Marked reports will be returned you with feedback and you should keep these as they and the feedback given on them should provide a basis for the reports you will have to write in subsequent years. ASSESSMENT OF PRACTICAL WORK The responsibility for handing your work in at the correct time is yours, and failure to do so will usually mean that your work will be marked for feedback pruposes, but that a mark of zero will be recorded. Exceptions to this rule will normally be made only for illness for which you have notified the School. If you do think you have another valid reason for missing the hand-in time, or for not attending the lab class in the first place, you should discuss this with the member of staff running the lab class or with the MO, Dr Carole Tucker. Each experiment and each report will be marked out of 20 in accordance with the scheme: 16 = very good performance; 14+ = first class level; 12 = competent 2i level performance; 10 = 2ii level; 8 = bare pass. Your module mark (see Undergraduate Handbook) will be made up as follows: Long report 33.3% Weekly lab diary marks 66.7%

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The experimental lab dairy notes of all experiments will be assessed with feedback weekly and the one long report will be assigned to you before the Easter vacation and given a % score. Your total module marks will normally be obtained by expressing the total marks you obtain during the session as a percentage of the total which you could have obtained during the session. Exceptions will normally be made in the cases of absence due to illness for which a medical certificate has been supplied; absence for an unavoidable reason of which you notified a member of staff, difficulty with an experiment for reasons which were not your responsibility and which you discussed with the demonstrator. As for PX1123, ATTENDANCE OF THE LABS IS COMPULSORY; absence requires a self assessment form to be submitted. Attendance is recorded and students are expected to sign out of the laboratory, if leaving before the end of the session. The submission of the Long Report is also a compulsory element of the course. REFRESHMENT ARRANGEMENTS Tea, coffee and snacks, will be available in the laboratory about halfway through the afternoon. Tea and coffee: Payment for these must be made at the beginning of the semester and will cover the whole semester. Prices will be announced at the first laboratory class. Snacks/chocolate: Payment individually at the time of purchase.

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SAFETY IN THE LABORATORY

Maintaining a safe working environment in the laboratory is paramount. The following points supplement those contained in "School of Physics Safety Regulations for Undergraduates", a copy of which was given to you when you registered in the School. 1. It is your responsibility to ensure that at all times you work in such a way as to ensure your

own safety and that of other persons in the laboratory. 2. The treatment of serious injuries must take precedence over all other action including the

containment or cleaning up of radioactive contamination. 3. None of the experiments in the laboratory is dangerous provided that normal practices are

followed. However, particular care should be exercised in those experiments involving cryogenic fluids, lasers and radioactive materials. Relevant safety information will be found in the scripts for these experiments and demonstrators will brief you prior to beginning the experiment.

4. If you are uncertain about any safety matter for any of the experiments, you MUST consult

a demonstrator. 5. All accidents must be reported to a laboratory supervisor or technician who will take the

necessary action. 6. After an accident a report form, which can be obtained from the technician, must be

completed and given to the laboratory supervisor. UNDERGRADUATE EXPERIMENT RISK ASSESSMENT

The experiments you will perform in the first year Physics Laboratory are relatively free of danger to health and safety. Nevertheless, an important element of your training in laboratory work will be to introduce you to the need to assess carefully any risks associated with a given experimental situation. As an aid towards this end, a sheet entitled Code of Practice for Teaching Laboratories follows. At the commencement of each experiment, you are asked to use the material on this sheet to arrive at a risk assessment of the experiment you are about to perform. A statement (which may, in some cases, be brief) of any risk(s) you perceive in the work should be recorded as an additional item in your laboratory diary account of the experiment.

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SCHOOL OF PHYSICS & ASTRONOMY: CODE OF PRACTICE FOR TEACHING LABORATORIES

Electricity Supplies to circuits using voltages greater than 25V ac or 60V dc should be

"hardwired" via plugs and sockets. Supplies of 25Vac, 60V dc or less should be connected using 4 mm plugs and insulated leads, the only exceptions being"breadboards". It is forbidden to open 13 A plugs.

Chemicals Before handling chemicals, the relevant Chemical Risk Assessment forms must

be obtained and read carefully. Radioactive Gloves must be worn and tweezers used when handling. Sources Lasers Never look directly into a laser beam. Experiments should be arranged to

minimise reflected beams. X-Rays The X-ray generators in the teaching laboratories are inherently safe, but the

safety procedures given must be strictly followed. Waste Disposal "Sharps", ie, hypodermic needles, broken glass and sharp metal pieces should

be put in the yellow containers provided. Photographic chemicals may be washed down the drain with plenty of water. Other chemicals should be given to the Technician or Demonstrator for disposal.

Liquid Nitrogen Great care should be taken when using as contact with skin can cause "cold

burns". Goggles and gloves must be worn when pouring. Natural Gas Only approved apparatus can be connected to the gas supplies and these

should be turned off when not in use. Compressed Air This can be dangerous if mis-handled and should be used with care. Any

flexible tubing connected must be secured to stop it moving when the supply is turned on.

Gas Cylinders Must be properly secured by clamping to a bench or placed in cylinder

stands. The correct regulators must be fitted. Machines When using machines, eg, lathe and drill, eye protection must be worn and

guards in place. Long hair and loose clothing especially ties should be secured so that they cannot be caught in rotating parts. Machines can only be used under supervision.

Hand Tools Care should be taken when using tools and hands kept away from the cutting

edges. Hot Plates Can cause burns. The temperature should be checked before handling. Ultrasonic Baths Avoid direct bodily contact with the bath when in operation.

Vacuum If glassware is evacuated, implosion guarding must be used in Equipment order to contain the glass in the event of an accident.

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II: EXPERIMENTS TIME TABLE AND LIST OF EXPERIMENTS

Week Experi- Title Page ment Spring Semester (PX1223)

1 – 5 12 Writing Long Reports 9 (see list) 13 Propogation of Sound in Gases 13

14 Magnetic Fields and Electric Currents 16 15

16 Variation of Resistance with Temperature Computer Data Logging and RC circuits

23 27

6 – 10 17 Optical Diffraction 34

(see list) 18 Measuring e/m for the Electron 38 19 X-rays 42 20 Microwaves 48 21 Computer Simulations and Analysis 55

11 Surprise!

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CHECKLIST • Read through notes on the experiment that you will be doing BEFORE coming to the practical

class. • Read carefully through any additional sections that might be useful in Section III – eg. use of

electronic equipment, statistics., and also the diary checklist given at the end of this manual. • Write a draft of the safety considerations that there might be associated with the practical, having read through the lab notes. • On turning up to the lab, listen carefully to any briefing that is given by your demonstrator: he/she

will give you tips on how to do the experiment as well as detailing any safety considerations relevant to your experiment.

• Write up the safety considerations and have your Risk Assessment signed off.. • Check that the size of any quantities that you have been asked to derive/calculate are sensible -

ie. are they the right order of magnitude? • Read through your account of your experiment before handing it in, checking that you have

included errors/error calculations, that you are quoting numbers to the correct number of significant figures and that you have included units.

• Staple any loose paper (eg. graphs, computer print-outs, questionnaires etc.) into your lab

book.

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Experiment 12: Writing Formal Reports At the end of PX1123 you wrote a Formal Report on one of the experiments you had performed during the semester. These will have been thoroughly marked and lots of feedback comments given. These will be handed back to you at the end of this session...... You will be required to write and submit another such report (after the Easter vacation) for PX1223 – it is expected that you will have taken on board the feedback given and can greatly improve upon your first attempt. This session is to assist with that process, so that you have a much clearer idea of what will be expected of your future formal reports (in 2nd year lab and your 3rd year long project). So part 1 is to read Appendix 1 - a reminder of the details given on report writing inthe PX1123 lab manual (it was obvious who hadn’t read this last time!). In part 2 you’ll be given a mock report – absolutely full of common mistakes. You are to go through this, mark it and make a list of all the errors. Your lab supervisor will then discuss these with you. Check them against the advice given in section 1. For part 3, you will be given 3 real reports to mark and rank in quality order. And finally, you should reread your own PX1123 report and understand the feedback you’ve been given. Ask for explanation – we want you to do a really good job next time! If you are uncertain as to how to use certain word-processing tools (for example an equation editor), this is a good opportunity to ask. APPENDIX 1. Advice on writing up formal reports of experiments and their results. AIM: to PRESENT the results of your work The person marking your full report is interested in your description of the experiment. They are not concerned with the actual measurements or quality of the results but are concerned with the way these are presented in the report. You should aim to present a clear, concise, report of the experiment you have performed, at a level able to be understood by a fellow 1st Year student, who does not have expert knowledge of your experiment. An example of a full report and further advice are given in PX1123 manual. Very importantly, your report must be original and not a copy of any part of the notes provided with the experiment. It should be a report of what you did; not of what you would like to have done or of what you think you should have done. That said, credit will be given for discussions on how one might extend and improve an experiment, and what might be done if the experiment were to be repeated. It is normal practise in writing scientific papers to omit all details of calculations, and you should also do this. Providing your report includes a statement of the basic theory which you used, together with a record of your experimental observations (summarized if appropriate) and the parameters which you obtain as a result of your calculations, it will be possible for anyone who so wishes to check the calculations you perform. The principles of report writing are simple: give the report a sensible structure; write in proper, concise English; use the past tense passive voice, for example "... the potentiometer was balanced

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...". The following structure is suggested. It is not mandatory, but you are strongly recommended to adopt it.

1) Follow the title with an abstract. Head this section “Abstract". • An abstract is a very brief (~50-100 words) synopsis of the experiment performed. An example is "The speed of sound in a gas has been measured using the standing wave cavity method for one gas (air) for a range of temperatures near room temperature and for gases of different molecular weights (air, argon, carbon dioxide) at room temperature. The speed in air near room temperature was found to be proportional to T½, where T is the gas temperature in Kelvin, and the ratio Cp/Cv for air, argon and carbon dioxide at room temperature was found to be 1.402 ± 0.003, 1.668 ± 0.003 and 1.300 ± 0.003 respectively".

2) Follow the abstract, on a separate page, with an introduction to the experiment.

Head this section “Introduction”. • Here, you should state the purpose of the experiment, and outline the principles upon which it was based (put some background physics in here). This section is often the most difficult to write. On many occasions it is convenient to draft all the rest of the report and write this last. Remember that the reader will, in general, not be as familiar with the subject matter as the author. Start with a brief general survey of the particular area of physics under investigation before plunging into details of the work performed.

• Important formulae and equations to be used later in the report can often, with advantage, be mentioned in the introduction as, by showing what quantities are to be measured, their presence helps in the understanding of the experiment. Formulae or equations should only be quoted at this stage. Derivations of formulae or equations should be given either by references to sources, for example text books, or in full in appendices. References should be given in the way described below.

3) Follow this with a description of the experimental procedure. Head this “Experimental Procedure”. • Write the experimental procedure as concisely as possible: give only the essentials, but do mention any difficulties you experienced and how they were overcome. Division of the description of the experimental procedure into sections, each one dealing with the measurement of one quantity, is often convenient. If the introduction to the experiment has been well designed this division will occur naturally. Relegate any matters which can be treated separately, such as proofs of formulae, to numbered appendices. Give references in the way described below. • All diagrams, graphs or figures should be labelled as figures. Give each a consecutive number (as Figure 1 etc.), a brief title and, where possible, a brief caption. Give each group or table of measurements a number (as Table 1 etc.) and a brief title, and use the numbers for reference from the text e.g. “the data in Figure 1 exhibits a straight….”

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4) Follow this section with the results of the experiment, discussion of them and comments. Head this “Results and discussion”.

• The result of the experiment can be stated quite briefly as "The value of X obtained was N + σ (N) UNITS". For example "The viscosity of water at 20°C was found to be (1.002 ± 0.001) x 10-3 N M-2 s". • Discussion of the result, or of measurements, method etc., can be cross-referenced by quoting the figure, table or report section numbers.

• Generally only show results in one form, usually either a table or a graph. For instance DON’T give a table of results and then show a graph of the exact same data. However if you have multiple sets of similar graphical results, then a summary table can be useful.

5) Follow this section with your conclusions. Head this “Conclusions”. • The conclusions should restate, concisely, what you have achieved including the results and associated uncertainties. Point the way forward for how you believe the experiment could be improved

6) Follow this section with references. Head this “References” or “Bibliography”. • The last section of the main body of the report is the bibliography, or list of references. It is essential to provide references. There are two main styles used (along with many subtle variations) to detail references. In the Harvard method, the name of the first author along with the year of publication is inserted in the text, with full details given, in alphabetical order, at the end of the document. The second style, favoured here is known as the Vancouver approach, is slightly different. At the point in your report at which you wish to make the reference, insert a number in square brackets, e.g. [1]. Numbers should start with [1] and be in the order in which they appear in the report. References should be given in the reference or bibliography section, and should be listed in the order in which they appear in the report. (Whatever referencing system you adopt, be consistent!) Where referencing a book, give the author list, title, publisher, place published, year and if relevant, page number eg. [1] H.D. Young, R.A. Freedman, University Physics, Pearson, San Francisco, 2004. In the case of a journal paper, give the author list, title of article, journal title, vol no., page no.s, year. e.g. [2] M.S. Bigelow, N.N. Lepeshkin & R.W. Boyd, “Ultra-slow and superluminal light propagation in solids at room temperature”, Journal of Physics: Condensed Matter, 16, pp.1321-1340, 2004.

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In the case of a webpage (note: use webpages carefully as information is sometimes incorrect), give title, institution responsible, web address, and very importantly the date on which the website was accessed eg. [3] “How Hearing Works”, HowStuffWorks inc., http://science.howstuffworks.com/hearing.htm, accessed 13th July 2008 7) Follow this section with any appendices. Head this “Appendices”. • Use the appendices to treat matters of detail which are not essential to the main part of the report, but that help to clarify or expand on points made. Give each appendix a different number to help cross referencing from other parts of the report and note that to be useful appendices must be mentioned in the main body of the report.

Health Warning: In subsequent years it may be necessary to develop this standard report layout to deal with complex experiments or series of experiments, so best get on top of it now.

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Experiment 13: Propogation of Sound in Gases. Note: This experiment is performed in the dark room. SAFETY ASPECTS: MAKE SURE THAT THE ROOM FAN IS SWITCHED TO EXTRACT AND IS WORKING. Outline The speed of sound is commonly used to refer specifically to the speed of sound waves in air, although the speed of sound can be measured in virtually any substance and will vary. The speed of sound in other gases will be dependent on the compressibility, density and temperature of the media. You will investigate these dependencies by studying the sound waves set up in various gases contained in a gas cavity. Experimental skills • Observation of longitudinal waves. • Understand the use of a microphone as an acoustic to electric transducer. • Hence using an oscilloscope to study non-electrical waves. • Careful use of gases and gas cylinders. Wider Applications • In dry air at 20°C, the speed of sound is 343 metres per second. This equates to 1,236

kilometres per hour, or about one kilometer in three seconds. The speed of sound in air is referred to as Mach 1 by aerospace professionals (i.e the ratio of air speed to local speed of sound =1).

• The physics of sound propogation, reflection and detection is used extensively for underwater locating (SONAR), robot navigation, atmospheric investigations and medical imaging (Ultrasound).

• The high speed of sound is responsible for the amusing "Donald Duck" voice which occurs when someone has breathed in helium from a balloon!

1. Introduction The speed of propagation of a sound disturbance in a gas depends upon the speed of the atoms or molecules that make up the gas, even though the movement of the atoms or molecules is localised. The r.m.s. speed of molecules of mass m in a gas at Kelvin-scale temperature T is given by;

2

1

21

2 3

=

mkT

c ,

where k is the Boltzmann constant. The sound is not propagated exactly at the speed < >c212 but at

21

3

γ

times it, where γ is the ratio of the principal heat capacities of the gas.

Thus

Csound =2

1

mkTγ

[1]

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Measurement of csound for known T and m therefore enables γ to be determined1. In this experiment the speed of sound in gaseous argon, air (mainly nitrogen) and carbon dioxide is measured by analysing the standing waves in a cavity. 2. Experiment 2.1 Apparatus The standing wave cavity is shown schematically in Figure 1.

Figure 1: Standing wave cavity

The loudspeaker, driven from an oscillator, directs sound into the tube; standing waves are obtained by adjustment of the piston and detected by the microphone insert at the end of the tube. The output from the microphone is amplified and displayed on the oscilloscope. Ensure that the amplifier is turned off when you have finished this experiment. Consider and write down the relationship between the length of the tube and the wavelength of sound for standing waves in closed and open tubes. Revise these expressions having considered this material using reference 2 or another source. Should you treat your equipment as having two closed ends or one open and one closed? Why? Show that the length of the tube L is related to the wavelength as L = ?/4, 3 ?/4, 5 ?/4, 7 ?/4

i.e. ( )4

12λ

−= nL , where n is an integer .

Note. The volume of sound coming from the speaker should be made as small as possible. Use the most sensitive Volts/Div setting on your oscilloscope. 2.2. Experimental procedure There may be traces of carbon dioxide in the tube from the previous experiment. This must be removed by pushing the piston in and out of the tube over its full travel several times.

Switch on the oscillator, and set it to give a sound at 1000 Hz. Find the approximate positions of the maxima in the signal amplitudes. Plot the signal amplitude as a function of piston position for all the accessible maxima (you will need to select a suitable step size). Now plot the piston position for each maximum on a graph and deduce the wavelength λ from the gradient. Calculate csound from the relation csound = fλ, where f is the frequency of the sound. Repeat the measurement for a number of other frequencies up to 5000 Hz. Consider whether there is any significant variation in your

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results, and attempt to account for it. Record the atmospheric temperature. Consider what affect the temperature might have on the measured speed of sound.

Repeat the experiment at one of the higher frequencies with the monatomic gas argon in the tube. Before attempting this, liaise with the demonstrator, who will arrange for the supply of the gas from the gas cylinder. Repeat the measurements at one frequency with carbon dioxide in the tube. Note any differences in the quality of the signal obtained. Why does this happen?

Use your results to calculate the value of γ, the ratio of the principal specfic heats of each of the three gases, from equation [1]. In equation [1], k = Boltzmann constant = 1.38 × 10-23 J K-1 T = temperature in Kelvin m = mass of one gas molecule i.e. relative molecular mass × 1.66 × 10-27 kg

The relative molecular masses of argon, nitrogen and carbon dioxide are 40.0, 28.0 and 44.0 respectively. Tabulate the values of γ you obtain, together with the values given by the kinetic theory of gases. 3. References 1 H.D. Young and R.A. Freedman, “University Physics”, Pearson, San Francisco, 2004, p547 2 Resnick & Walker, “Principles of Physics”, Wiley edition 9, p457.

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Experiment 14: Magnetic Fields and Electric Currents.

Equipment List: Current balance, rheostat (a coil of wire with a slider used to vary its effective resistance), Weir p.s.u., multi-meter (rated to 10 A), small magnetic compass, A4 paper.

Safety. The current balance may spark. The resistor can get VERY hot over time.

Outline The shape of the magnetic field lines in the vicinity of two separated permanent magnets and around a current carrying wire is investigated using small magnetic compasses. The force on a current carrying wire passing through the magnetic field of permanent magnets is then investigated using a “current balance” and used to obtain a value for the size of the magnetic field. The experiment illustrates the properties introduction to magnetic materials and essential concepts of electromagnetic theory.

Experimental skills • Make and record measurements of magnetic field lines. • Familiarity with the magnitude of magnetic fields generated by electrical currents and permanent

magnets. • Experience of the effect of stray magnetic fields in a laboratory environment. • Application of vector cross products to real situations. • Use of ballast resistor to limit current flowing in circuit.

Historical perspective and wider applications Magnetic materials: the use of lodestone as a crude magnetic compass dates to ~1000 BC. Electromagnetism: In 1819 in Copenhagen Hans Oersted discovered, almost by accident, that a compass needle can be influenced by a nearby electrical current. This was the birth of electromagnetism, one of the most important fields in both science and engineering, with profound influence on modern life: • Michael Faraday discovered electromagnetic induction and developed the idea of a field for

dealing action at a distance effects. • These ideas led to delopment of the dynamo, motor and transformer. • James Clerk Maxwell put the field ideas into mathematical form and predicted electromagnetic

waves. • Einstein’s consideration of the need for relative motion led to the theory of relativity.

1. Introduction Magnetic fields can arise from magnetic materials and from moving charges. This experiment is concerned with examining both such fields and also the forces resulting from the interaction between magnetic fields and moving charges (due to a current flowing through a wire).

1.1 Magnetic fields Magnetic fields are vectors and therefore have both a direction and a magnitude (or strength). They are produced by magnetic objects or by moving charges. The oldest known magnetic field is that due to the Earth and this leads to the concept of poles and the first way of defining the direction of the field. , i.e. a “North pole” will point to the Earth’s North pole (which since opposite poles attract magnetically must itself be a South pole).

• The direction of a magnetic field is defined to be that in which a North pole will move.

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• Magnetic compasses point in the direction of a magnetic field, i.e. towards a magnetic south pole.

Magnetic fields can vary wildly in both magnitude and direction as a function of position, are therefore mathematically complex, and are often visualised by way of “field lines”. These are constructed by using arrows to indicate the direction of the field at various points and then connected by lines. The number of lines used must be limited and this is done in such a way that the density of the lines in the vicinity of a point gives an indication of the relative strength of the field. An example, representing a bar magnet, is shown in Figure 1. The permanent magnets used here are similar to the one shown except that their poles are wider than their length.

Figure 1: Magnetic field lines in the vicinity of a bar magnet [1]

Figure 1 also hints at another important property of magnetic field lines. Unlike electric or gravitational field lines they form loops. This relates to the fact that there is no such thing as a magnetic monopole.

1.2 Electromagnetic theory (and vector cross products) Electromagnetic theory gives the magnetic force, F, exerted on a charge, q, moving with velocity, v, in a magnetic field as

F = qv x B (N) [1]

At the same time the magnetic field generated by a point charge moving with velocity v is

[ ]rvB ×=2

0

4 r

q

π

µ (tesla, T) [2]

where r is the vector from the point charge to the point at which the field is determined and µ0 is the permittivity of free space (µ0 = 4p x 10-7 H/m or 1.26 x 10-6 TmA-1). These definitions are given as vector cross products, so although students may be more familiar with the use of Flemings left and right hand rule for determining directions here it makes more sense to use the more general rules for dealing with vectors. The case is illustrated for two vectors a and b is shown in Figure 2.

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a

b

a x b = c

b x a = -c

?

z direction

Figure 2: The cross products of two vectors a and b separated by an angle ?. The resultants are in a direction perpendicular to the plane containing both a and b.

For the cross product c = a × b the direction is perpendicular to the plane formed by a and b and its direction is given by the Right Hand Rule*: • Imagine your right hand pointing along a. • Curl the fingers around from a to b. • The thumb then points in the direction of c.

* From this the coordinate system being use is said to be right handed. As drawn above, a × b = c is in direction = +z whereas b × a = - c is in the negative z direction. (In a left handed system following a left handed rule the directions are reversed). Using this rule, and bearing in mind that the move charges in this experiment will always be negatively charged electrons, equations 1 and 2 can be used to determine the direction of both force and magnetic field vectors. Note: Ultimately these two models for magnetic fields, poles and flowing currents, are identical and equivalent and the magnetic fields produced by magnetic materials originate in microscopic currents flowing cooperatively. The magnetic pole model is therefore a simplistic viewpoint but one that is very useful in many circumstances. Both approaches will be employed here.

1.3 Charges moving in a wire The above descriptions for individual charges whilst useful for considering the direction of force and field vectors requires development for the situation here where there are many moving charges (electrons) and all are confined to a metallic wire. For a conductor carrying a current in a magnetic field in the case where the current, I and field, B are perpendicular the force on the wire is given by

F = BIL (N) [3]

where L is the length of the wire in the field. This comes from a consideration of the number and velocity of charges experiencing the magnetic field and is derived in the lecture courses and in Young and Freeman. Somewhat similar considerations can be applied to the magnitude of the magnetic field around a straight conductor. The field lines in this case are circles concentric with the wire and decrease with distance r from the wire. For an infinitely long conductor the magnitude of the field is given by:

rI

Bπ

µ2

0= (tesla, T) [4]

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Magnetic field lines due to a current in a wire are shown in Figure 3.

currentcarryingwire

magneticfieldlines

Figure 3. Magnetic field lines surrounding a current carrying wire. For the direction of the field lines shown the current is in a direction out of the page.

2. Experimental

2.1 Apparatus (the current balance) The equipment, shown part assembled in Figure 4, consists of a copper frame (scribed on one side) which balances on two pivot edges. A break in the frame, in the region of the pointer, ensures that any current flowing between the pivots only passes through one “arm” of the frame. The pointer can be positioned in the opening of a support that restricts its movement. The current carrying arm is placed in the magnetic field centrally between the poles of strong permanent magnets mounted on mild steel yokes. With this arrangement, the current, magnetic field and movement of the wire are all at right angles and so equation 3 applies.

Figure 4: Frame mounted on centrally positioned pivot edges. The pointer is to the left and is shown within the support. Current flows only through the arm on the right, passing between the poles of permanent magnets.

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Electrical circuit: The copper frame has a very low resistance (~0.2 O) so to protect the power supply unit and the equipment (from high currents) a ballast resistance of ~5 O should be placed in series with the frame. The variable resistor (rheostat) provided is a suitable ballast (in terms of resistance value and current capacity). The rheostat has three terminals and a maximum resistance of ~10 O. To obtain a resistance of ~5O simply move the top slider half way along the coil and make sure to use the top and one of the bottom connectors. The power supply unit (dc output) and an ammeter set to its 10 A range and also in-series completes the circuit.

When required use the dial on the poser supply unit to set the current.

IMPORTANT: Currents must not be allowed to exceed 2.5 A and reduce the current to zero between measurements.

Magnets: When making calculations it will be assumed that the “magnets” are exactly 5 cm in length, have no “edge effects”. No “edge effects” implies that the magnetic field confined to the region directly between the poles - in reality it spreads a little. This is addressed again in section 2.2. Weights: In this experiment, small pieces of photocopier paper (cut up using scissors) will be used. A figure of merit for paper is its areal mass density and the photocopier paper used by the School is indicated to be 80 g/m2. Measurements show that this figure is accurate to +/- 1% and so the areal density should be written as 80.0 +/- 0.8 g/m2. This accuracy is more than sufficient for the purposes of this experiment. Since the wire frame balances on a pivot, forces on the frame should be considered as moments. However if masses are added on the same section of the frame that passes through the magnets, the distance from the points of application of the force to the pivot is the same and it is sufficient to only consider forces. Field line measurements: Early experiments examine the shape of (permanent) magnetic field lines and small magnetic compasses are used for this purpose.

2.2 The magnetic field lines associated with permanent magnets The nature of the magnetic field surrounding a single permanent bar magnet with a similar geometry to that used in this experiment is shown in figure 1. This part of the experiment examines the more complicated case of: (i) two such magnets separated by a fixed gap; (ii) two such magnets separated by the same extent but mounted on a “U” shaped yoke. Set up • On a fresh piece of A4 paper place the two magnets, centrally and with N pole facing S so that

they attract first of all separated by the wooden block. The wooden block is not magnetic and so has no effect on observations).

• Trace around the magnets so that they can be re-positioned if moved accidentally. Experiment • Use the small compass to determine the direction of the field lines* in the vicinity of the magnets:

find the direction of the field line at a point, draw an arrow in the position of the compass, move the compass along in the direction of the field line and repeat. Concentrate on one side of the magnet and take enough measurements to illustrate symmetry and to generate a reasonably accurate impression of the field lines (as in Figure 1).

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• Repeat the process for the same magnets separated by a “U” shaped yoke (the magnets should still be oriented N-S and the wooden block should be removed).

• Describe and attempt to account for the difference between the two cases. *A useful point to note: after being disturbed the compass needle exhibits a damped oscillation, whose frequency increases with field strength.

2.3 Oersted’s experiment (A classic experiment of physics) Reminder: Oersted’s experiment, that started the field of electromagnetism, was simply the observation that currents travelling through wires affected a magnetic compass in its vicinity. Here the effect will be used to confirm the cross product expression given in equation 2. Set up • The equipment should be set up as shown in Figure 4, although the magnets are not required at

this stage and it is not important for the frame to be balanced, it can be held horizontal using the support (shown on the left).

• Connect the power supply unit using the dc output: Use red wires to connect the current balance to the positive output and black wires to the negative output (this will help when determining the direction of charge flow) and pass the current through an ammeter on its 10 A range.

Experiment • Place the small compass close to the frame (as close as possible without touching) and confirm,

such as by increasing the current to 2.5 A and then decreasing it again in different positions around the wire, that the current has an effect on the compass. This, in essence, was Oersted’s experiment. Take care, the wire will spark.

Whilst a movement of the compass needle due to the current in the wire should be obvious it is true that the effect is weak. Most notably the contribution due to the current is competing with the Earth’s magnetic field (which varies with position but is in the range 30-60 µT) and with that due to the steel in the bench system.

• Use estimates and observations to decide the origin of the largest contribution to the field experienced by a compass when it is as close as possible to the wire carrying a current of 2.5 A.

• Passing a current of 2.5 A through the wire for short periods, and with reference to Figure 2, use the compass to determine the direction of the magnetic field. Confirm, through consideration of the direction of current/charge flow, that the direction is as predicted by equation 2. (Demonstrators will expect to see a suitably labelled diagram here).

2.4 Investigation of a force on a current carrying wire in a magnetic field The current I (A) and length L (m) of wire in the field can be varied independently and the magnetic force F (N) measured by balancing it against the force due to known masses in the gravitational field. The magnetic field B (T) is determined by the strength of the permanent magnets and their separation and has a constant value that is measured in this experiment. Once the magnetic field strength has been found the apparatus is used as a mass balance to measure (relatively small) masses. Set up

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• Connect the voltage source, the rheostat, the ammeter and the balance in series. The rheostat is a coil of wire with a slider used to vary its effective resistance. It is a useful way of controlling current in this experiment.

• The next objective is to balance the frame with no magnetic forces acting on it. To aid this one side of each frame has been finely scribed. Locate the scribed grooves on the balance with the pointer between the balance indicator (this will limit the movement of the frame). Finely balance the frame by moving the small metal rider along the frame (best done with tweezers, but bear in mibd they are magnetic).

• Position the magnet so that the frame lies centrally between the “magnet’s” pole-pieces. • Pass a current through the frame, ensuring that the current is such that the arm is raised. This

upwards force will later be counterbalanced by weights placed on the same section of the arm that passes through the magnet.

Experiment • Cut out a square or rectangle of paper, measure its dimensions and place it on the balance. • Increase the current until the beam is balanced. • Repeat the previous steps using different or additional areas of paper. • Plot a suitable graph and use it to show that F is proportional to I and to calculate the magnetic

field, B for the magnets used. Note: Clearly it is important that the frame and rider do not move during the course of the experiment. If they move or are suspected to have moved it will be necessary to rebalance the system with no masses and no current flowing. References 1. http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html (accessed 2/11/10)

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Experiment 15: Variation of Resistance with Temperature. Safety Aspects: In this experiment you will use the cryogen liquid nitrogen (boiling point 77.3K). Please ensure that you read the safety precautions, write a risk assessment AND seek the assistance of a demonstrator before using this. SAFETY PRECAUTIONS IN THE HANDLING OF LIQUID NlTROGEN Avoid contact with the fluid, and therefore avoid splashing of the liquid when transferring it from one vessel to another. Remember that when filling a "warm" dewar, excessive boil-off occurs and therefore a slow and careful transfer is necessary. Do not permit the liquid to become trapped in an unvented system. If you do not wear spectacles, safety glasses (which are provided) must be worn when liquid nitrogen is being transferred from one vessel to another. FIRST AID If liquid nitrogen contacts the skin, flush the affected area with water. If any visible ''burn" results contact a member of staff. Outline All materials can be broadly separated into 3 classes, according to their electrical resistance; metals, insulators and semiconductors. This resistance to the flow of charge is temperature dependent but the dependence is not the same for all material classes, because of the physical processes involved. In this experiment you will determine the behaviour of electrical resistance as a function of temperature for a metal and a semiconductor. You will confirm the linearity or otherwise of these behaviours. Experimental skills • Ability to keep a clear head and organize a one-off experiment, paying careful attention to safety

aspects. • Make and record simultaneous measurements of a number of time-varying quantities. • Determine realistic errors in these quantities and combine them. • Gain experience of liquid cryogens. • Fit measured data to linear, polynomial and logarithmic expressions. Wider Applications • Many branches of physics and its applications involve the study and use of materials at

cryogenic temperatures (those below ~ 150K). By understanding the temperature dependence of material behaviour, we can use it to our advantage.

• Modern imaging and communication systems rely on the sensitive, noiseless and reproducible detection and transfer of electrical information. This is often achived by using cooled semiconductor devices.

• Some materials become superconducting at cryogenic temperatures (i.e a temperature somewhat above absolute zero). This phenomenon has found application in Medical imaging (MRI scanners depend on the huge magnetic fields achievable only by using superconducting

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coils); Astronomical imaging (superconducting detectors are used to count 13 billion year old photons) and transport (MAGLEV trains).

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1. Introduction In this experiment you will investigate the variation of the resistance of: 1) a semiconductor (a thermistor); 2) a metal (copper) in the temperature range from ~ 120 - 290 K. For a metal the following equation [1] describes the linear behaviour of resistance R with temperature T. R(T)= R273(1 + α(T-273)) , [1] Where R(T) is the resistance at temperature T (in Kelvin), R273 is the resistance at 273K andα is a constant known as the temperature coefficient of resistance, which depends on the material being considered and will vary slightly with the reference temperature (273K here). However the behaviour may be more closely described by a 2nd order polynomial fit. RT = R273 1 + α(T-273) + β(T-273) 2, [2] where β is another constant. For a typical intrinsic semiconductor the electrical resistance obeys an exponential relationship with temperature. It takes the form of equation [3] . RT = a eb/T , [3] where RT is the resistance at T and a and b are constants. By using equations [1], [2] and [3], you are to find suitable graphical ways to verify or disprove these relationships. You may use Excel (or another plotting package familiar to you) to plot your data, BUT remember to take care with axes, apply suitable error bars and think about what your results mean. 2. Experiment 2.1 Apparatus The metal you will test is in the form of a coil of fine wire. The semiconductor is a thermistor. Both of these are attached to the top of a copper rod. They are held in good thermal contact with it by a low-temperature varnish. The temperature of the specimens can be reduced by immersing the copper rod to various depths in liquid nitrogen, which boils at 77.3 K. The liquid nitrogen is poured into a Dewar flask contained in the box which supports the copper-rod assembly. The liquid-nitrogen level is gradually increased by adding liquid nitrogen through the funnel.

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An insulating cap is provided which, when placed over the top of the rod, thermally isolates the specimens from the surroundings and allows their temperature to fall to a value determined by the depth of immersion of the rod in the liquid nitrogen. The temperature of the specimens is measured with a thermocouple. This consists of two junctions of dissimilar metals arranged as shown in Figure 1. If the two junctions are at different temperatures an e.m.f. is generated which, to a good approximation, is proportional to the temperature difference between the two junctions. By calibrating such a thermocouple, temperature differences can be determined by voltage measurements and these can be used to measure temperature if one standard junction is held at a well-defined fixed temperature.

Figure 1: Representation of back-to-back thermocouple junctions and circuit

In this experiment we use a copper-constantan thermocouple. One junction of this is embedded with the specimens in the varnish; the other, the standard, is kept at 77.3 K by immersion in liquid nitrogen contained in a separate Dewar flask. You will calibrate the thermocouple with the standard junction in liquid nitrogen while that attached to the metal rod remains at room temperature. The resistances of the copper and thermistor are read from multimeters suitably connected. The voltage across the thermocouple is also read by a multimeter. Ensure you can read all 3 scales simultaneously. 2.2 Calibration of the thermocouple Connect a multimeter to the appropriate thermocouple terminals on top of the rod. Immerse the free junction in liquid nitrogen and record a voltage. Take another voltage reading when the junction is at room temperature. You can now calibrate the thermocouple scale by assuming that the voltage is linearly related to temperature difference. (This is not strictly true but will suffice for our purposes.) Check your calibration with a demonstrator and ensure that you know how to use the thermocouple as a thermometer for the rest of the experiment. 2.3 Resistance measurements The magnitudes of the coil and thermistor resistances will be determined using multimeters set to the ohms range.

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Measure RC (the resistance of the copper coil) and RT h (the resistance of the thermistor) at room temperature. Place the insulating cap on top of the rod and start to add liquid nitrogen through the funnel. Note the readings on the 3 multimeters (thermocouple voltage, Rc and RT h). Gradually add more liquid nitrogen and repeat .The object of the experiment is to obtain as many measurements of Rc and RT as possible over as wide a temperature range as possible. Remember to ensure that you have a simple diagram of your apparatus that would allow you to set the experiment up again. Experimental Notes

• You must work quickly and efficiently if you are to obtain sufficient experimental points on the graphs

• Handle the Dewar flasks carefully. • DO NOT touch the copper rod when it has been immersed in liquid nitrogen. If you do, you

may freeze to the cold metal and give yourself a severe burn • You will find that there will be little change in temperature of the coil and the thermistor

when liquid nitrogen is added initially, but take care not to add too much liquid nitrogen at any one time or a large temperature drop may result. Once the rod has been cooled, it is not easy to raise the temperature again in the course of the experiment. This is a one hit expereiment!

• The lowest temperature you are likely to reach will be at best ~ 120 K. • Make notes in your lab diaries of anything that happens during the experiment, e.g. where

you note a change of range on the multimeter. • Make a note in your lab diary of the specific pieces of equipment that you have used.

3. Data analysis Plot suitable graphs of your data and investigate the validity of equations [1] and [2] for the metal and equation [3] for the thermistor. Finding values of α, β , a and b. You may use a computer package (Excel is recommended) to fit the equations but be careful to check your axes, show error information and quote gradients and results to a sensible number of significant figures. Does the variation of resistance in a metal vary linearly with temperature? Which equation gives the best fit to the data? What do you notice about the variation for a semiconductor? Is the exponential fit of equation [3] good enough? How might the experiment, errors in the data, or your experimental method be improved?

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Experiment 16: Computer Data Acquisition (and RC Circuits) Use of computers in this experiment Due to multiple users of these computers, do not save copies of your work onto the PC hard-drive. Ensure that you obtain hard copies of your data as you go along, by using the printer.

Outline Although most experiments in the first year laboratory involve taking measurements by hand, the use of data-loggers, often run by computers, is ubiquitous at research level. This experiment demonstrates both the advantages and potential pitfalls when one uses a computer to digitally capture an oscillating (analogue) voltage produced by a signal generator. “Under sampling” of the signal, which leads to “aliasing” is revealed by use of Fourier transformations of the data to reveal the frequency components. Finally the system is used to measure the discharge of a resistance-capacitor (RC) circuit and determine its “time constant”. Experimental skills • Basic use of data loggers and signal generators. • Awareness of digital signal processing effects: (under) sampling leading to aliasing. • Use of analysis functions built into software. • A very basic introduction to the use of Fourier transformations in the analysis of periodic

functions. • Very simple wiring (of an RC circuit). Wider Applications • Computer based data acquisition and associated signal processing is everywhere! • Fourier transforms are not restricted in signal processing/analysis but appear in many fields of

physics. For example, undergraduates are often taught the mathematics of Fourier transforms in optics courses. Closely related, and performed in 1st year laboratory, x-ray diffraction patterns can be considered to represent the 3D Fourier transform of crystal lattices. You will not be taught the maths of Fourier theory yet, but an appreciation of its uses will be advantageous in future discussions.

• RC circuits are widely used for example to modify analogue voltage signals. A later experiment considers their use in rectifying circuits which convert ac voltages to dc - the RC component appears in the final “smoothing” of the signal.

1. Introduction Note: A data logger will be taken to be a device that records measurements to file usually as a function of time. Computer systems may also control an experiment by both setting an independent variable and measuring a dependent variable, but can equally act simply as a data logger. Performing experiments and taking data automatically via use of data loggers or computer controlled equipment is almost universal at research level. The advantages of computers are most obvious in the cases where there are large amounts of data to be handled (acquired, processed and stored) and/or where values change at a high rate. However, there are significant disadvantages. A major

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one is in the perception by many people that all errors disappear when measurements are made by computers - they most definitely do not! This is perhaps borne by the inscrutability of computers - it can very difficult to figure out what they are doing and how they arrive at their answers. The above limitations would be acceptable if our undergraduate courses were aimed at training operators, but they are not. As a scientist it is necessary to understand and trust the results of experiments and to do this it is necessary to understand the equipment and their associated limitations.

Apparatus PC, Pocket CASSY with U,I sensors, Signal generator, Cyclon cell, 5000µF capacitor, 20 kO resistor.

1.1 Data acquisition Systems used to acquiring data in the form of software files can superficially come in many forms but ultimately are all very similar. Whilst not attempting to be exhaustive, systems have the following features: • Starting with the parameter to be measured; this may come in many different forms, e.g.

temperature, displacement speed, light intensity etc etc. However, in all cases a “transducer” is used to measure the parameter and convert it with a well known conversion factor, into a voltage.

• A voltage from a transducer that varies continuously with time is known as an analogue signal. • This analogue signal must be converted to a digital signal in order to be read by and make sense

to a computer and this is achieved by an analogue to digital converter (A? D converter). • A? D converters output values with a fixed number, n of bits and consequently have a fixed

resolution. The CASSY system uses a 12 bit converter and so the signal for a particular range is split into 2n values. A voltage range of +/-3 V therefore has a resolution of 6/212 ~ 1.46 mV (this is what produces steps in data).

• The time axis too will be subject to limitations. Data is collected or “sampled” at set time intervals. Acquisition systems can be limited by the minimum time interval allowed and/or by the maximum number of points that can taken. Here a maximum of 16000 points can be collected with a minimum sampling interval of 100 µs

1.2 Controlling experiments In the set up used in this experiment the computer is simply involved in passively taking data, i.e. in data logging. However, the CASSY system can also control experiments, for example by setting a variable and measuring a dependent variable. There are a number of experiments run under the CASSY system in the second year laboratories.

1.3 Sampling data The importance of the sampling interval is illustrated in Figure 1 in which an analogue sinusoidal signal is sampled at an interval that is slightly less than the period of oscillation. Sampled in this way the data would appear on a computer screen as a succession of points that bear no relation to the original data. Clearly a good representation of the original data simply

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requires a large (or sufficient) number of points per period of oscillation - or equivalently a sample frequency much higher than the signal frequency.

-1

-0.5

0

0.5

1

1.5

-1 -0.5 0 0.5 1

time /s

sign

al

Figure 1. A sinusoidal signal with a period of 0.5 s (frequency of 2 Hz) sampled (indicated by

dots) at a sampling interval of ~0.32 s (sampling frequency, fsample ~3.1 Hz).

Of interest in this experiment is in identifying the sampling frequency sufficient to allow the extraction of meaningful information and what happens when the sampling frequency is insufficient, i.e. the signal is “under sampled”.

It turns out that to extract the frequency of a signal (see next section) the minimum requirement is a sampling frequency twice the signal frequency - a useful “rule of thumb” for experimentalists. However, to obtain an accurate signal shape much higher sampling frequencies are required.

1.4 Fourier transforms and aliasing Fourier transforms (FTs) will be used as a “black-box” signal processing/analysis tool in this experiment. The mathematics of how they work will come later in taught modules. This section only aims to give an introduction. The term “Fast Fourier Transform (FFT)” will be seen and this refers to how the transform is performed, here Fourier and Fast Fourier Transforms will be taken to mean the same thing. In Figure 8.1 the signal is displayed in the “time (t) domain”, i.e. the signal is plotted against time on the x-axis. However, it is also possible to represent the signal in the “frequency (1/t or f) domain” i.e. the x-axis is in terms of frequency. Since the signal in figure 1 is composed of a function oscillating with a single frequency when represented in the frequency domain it would be expected to be a single peak at this frequency. “Fourier transform” is the term used to describe both the process of converting data between domains and the new representation of the data. The power of FTs in signal processing arises from its ability to take a complicated signal made by the sum of a range of frequencies and split them into its components. For example, taking a musical chord, a FT would display the notes that make it up.

A particular effect of under-sampling a signal that will be examined using FTs is that of aliasing. Aliasing generally refers to effects that result in different signals becoming indistinguishable, i.e. “aliases” of one another.

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2. Experiment The practice of computer data acquisition is addressed initially by examining the importance of the sampling frequency in measuring sinusoidal signals from a signal generator, aided by the use of Fourier transforms. This is followed by a study of the discharge of a capacitor through a resistor

2.1 Computer and software • If not already started, turn on the computer. As the computer is not networked, log on using the

“student” account (no password required). • To start the CASSY software, from the “start” button click on “programs”, “CASSY Lab” and

then “CASSY Lab” again. • You should now see the “Settings” window. An icon representing the “pocket CASSY”

interface box and its U, I sensor should be visible. It is from here that the settings associated with the acquisition and display of data are set.

• Since we want to use the U input, which is on the left hand side of the sensor, left-click on this part of the icon. Three windows will now appear: “sensor input settings”, “measuring parameters” and “voltage U1”. The changes that need to be made from the default settings are described below:

sensor input settings - the default measuring range is +/- 10V, this needs to be changed to +/-3V This range is large enough to cope with the signal generator output and battery voltage (~2 V) and has the required (millivolt) precision. measuring parameters - the “measuring interval” (dt), the time between points and the “number” of points (n) are set in this window. The product of these two values (n.dt) gives the total measuring time and this value is also displayed. Note: that if n is not specified the system will collect data until instructed to stop and the graph will keep resetting the time axis to accommodate the readings. voltage U1 - simply gives a reading of the voltage value, both as a digital value and a pointer.

2.2 Examination of sinusoidal waveforms Methodology Setting up for a measurement • Connect the signal generator, from the 50O output, across the “U” input connections of the

small CASSY interface box. • A signal generator amplitude of ~1 V and zero dc offset is required. This is conveniently

achieved by: selecting a square wave from the signal generator (SG); setting a frequency fSG ~ 5Hz, making sure DC offset is turned off (button out), then adjusting the output level and observing the pointer.

• Select a sinusoidal output on the signal generator. • Before acquiring data. In the “measuring parameters” window (if you can’t see this click the

“toolbox” icon) set measuring interval as dt = 10 ms (fsample = 100 Hz) and n = 250 (measuring time 2.5s). These parameters will remain unchanged for the following examination of sinusoidal waveforms.

• Click the “stopwatch” icon to start taking measurements. Right click on the y axis of the displayed graph and choose an appropriate scale. To take another set of data, click on the clock icon again, collection will start as soon as save data?: “no” is clicked (and yes this is slightly quirky).

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Survey measurements at 5, 20 and 100 Hz • Qualitatively describe the recorded fSG = 5 Hz signal (20 measurements per cycle) - are there

sufficient samples for the recorded data to be a good representation of a sinusoidal waveform of constant amplitude and frequency?

• Repeat for fSG = 20 Hz (5 measurements per cycle) and 100 Hz (1 per cycle) signals. • Print out the graph (not the data table) for the final frequency, fSG = 100 Hz.

The above should have made the point that data acquisition can go wrong, the next task is to examine what is happening more closely.

Quantitative frequency measurements (fSG =10-100 Hz) • Acquire data at a frequency of fSG =10 Hz. • As accurately as possible, determine the period and so the frequency of the signal from the

graph, fgraph, (right click on the graph and “display coordinates” of the cross-hair”). Note from here-on measurements are being made so errors are required.

(Fast) Fourier Transforms will now be introduced. To set these up: • Click on the toolbox icon and select “Parameter/Formula/FFT”. • Click on the “New Quantity” button and select “Fast Fourier Transform” half way down the

settings box. • A “Frequency spectrum” tab should have appeared above the data table on the left of the

screen, click on this tab to show the FT. Since the signal is a single frequency a single peak should have appeared.

• Measure this frequency, fFT and compare with fSG and fgraph.

For the first measurement performed the frequency will be determined by examining both the V(t) data and its FT. Subsequently only the FT method will be used. • Returning to 10 Hz, acquire a measurement. From the V(t) data determine the period and so

the frequency (with an estimate of its error). • Look at the Fourier transform (frequency spectrum). As described above there should be a

single peak corresponding to the signal frequency (fFT). Use the cursors to find the frequency (with an estimate of its error).

• Compare the three frequencies: as set on signal generator (fSG), measured from signal versus time graph (ft), measured from FT (fFT).

• Now acquire data for fSG = 20 to 100 Hz in 10 Hz steps. Look at and briefly describe the Vs(t) data at each frequency but only use the FT to measure the frequency, noting your values in a table.

• Plot a graph of fFT/fSG versus fSG. • Comment on the meaning of this graph (e.g. the “rule of thumb” implies that fFT = fSG for fsample

= 2fSG should be observed, what is the agreement between fFT and fSG at low fSG?).

Note that as a result of the sampling frequency a signal may have a different, lower (measured) frequency assigned to it; it therefore has an “alias”.

2.3 Measurement of the discharge of a capacitor through a resistor.

This is a convenient experiment to perform both as an example of computer data acquisition and as preparation for later experiments with electrical circuits.

Resistor-capacitor circuits (a reminder)

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A capacitor is a device which may store electrical charge. Equal positive, +Q, and negative, -Q, charges are held on conductors inside the capacitor, so that there is overall charge neutrality. The greater the charge Q that is stored in the capacitor, the greater is the potential difference V between its two terminals: C/QV = [1]

where the constant C is called the capacitance of the capacitor.

By connecting a capacitor across a battery, charge will flow onto the conducting plates of the capacitor until the voltage across the capacitor equals that across the battery, so preventing any further charge flow. A schematic of the circuit used in this experiment is shown in Figure 2. The rate at which the capacitor charges depends on the battery voltage V0, the capacitance C, and the resistance R of the circuit.

Figure 2: Circuit arrangement

The voltage across the capacitor, charging from 0 V and starting at t = 0 s is given by the equation

)e(VV RC/t−−= 10 [2]

A charged capacitor may be discharged by connecting a wire across its terminals, and the rate of discharge again depends on V, C and the resistance R. The voltage across the capacitor, discharging from V0 and starting at t = 0 s is given by the equation:

RC/teVV −= 0 [3]

The quantity RC must have the dimensions of time if equations [1] and [2] are to be correct. The product RC is known as the "time constant" of the circuit, the higher the RC the slower the circuit charges and discharges.

In the experiment the charging of a capacitor will be used to determine suitable parameters for the CASSY system, but only the discharge of the capacitor will be analysed. Equation [3] for the capacitor discharge may be written by taking natural logs as: RCtVV /ln)ln( 0 −= [4]

Thus either C or R can be determined graphically if the other quantity is known. (τ is known as the time constant and gives us a measure of the rate of charging of a particular RC combination.)

Experimental apparatus and technique • Be very careful to ensure that you do not short circuit either the battery or the

(charged) capacitor; this can result in the flow of very large currents.

R

C

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• You are provided with capacitors and resistors whose values have a quoted tolerance of ±10% and ±5% respectively.

• The 5000µF capacitor and 20kO resistor will be measured. For this combination calculate the expected time constant, τ = CR.

Initial measurements • Before making up the circuit in Figure 8.2, use the system to measure the e.m.f. of the battery. • In the same way check that the capacitor is fully discharged. If it isn’t connect it to a 1 kO

resistor until it is fully discharged. • Without making the connection to the battery, set up the circuit in Figure 2 with the data logger

connected across the capacitance. • With the measuring (sample) interval still set to 100 ms and the number of points not specified,

start taking data then make the connection to the battery. Whilst data is being taken: by right clicking on the y-axis set a suitable scale (the x (time) axis should re-scale itself).

• With a limit of 125* on the number of data points decide on suitable measuring interval and time to acquire the discharge data. Ensure the capacitor is fully charged before acquiring these data.

* The data will be exported to EXCEL and 100 points is more convenient than thousands. Measuring the capacitor discharge • Set the measurement parameters determined above. • When ready, start data acquisition then start the discharge by removing both connections from

the battery (to remove it entirely from the circuit) before shorting them together. • Print out the graph, but do not print out the table. Analysing the data As is often the case, although there are a lot of analysis functions built into the data acquisition software, they do not necessarily the best tools to use. Here, although the software will find the logarithm of data it only uses log10 whereas equation [4] requires loge. Although the conversion is not difficult it is easier and more instructive to export the data. (Another good reason to export data is that the graphs produced by CASSY are not of good enough quality for formal reports.) • Open EXCEL • Right click on the data table and copy/paste it to EXCEL. • Use the graph function on EXCEL to make an appropriate graph in order to determine the time

constant (RC). • There is now a problem. Although EXCEL will give a quality of fit number it does not give an

error; and for any value determined in an experiment to have meaning there must be an associated error.

• To determine errors print out the graph and do it by hand in the usual way. • Compare the time constant and V0 values with expectations. • Use a multi-meter to determine an accurate resistance and use this to calculate the best value for

the capacitance. Within tolerance, are your values in agreement with those quoted?

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Experiment 17: Optical Diffraction Safety Aspects: You must take great care when using the laser to avoid damage to your eyes. In no circumstances must you look along the main beam. You must also take care that specularly reflected beams do not enter your eye when you are adjusting the various components. Check with a demonstrator before starting the experiment. Before coming to the lab, remind yourself about optical diffraction. Use an A level reference or read some of Chapter 36 (p990) of The Wiley Plus “Principles of Physics”. Outline In optics, Fraunhofer (or far-field) diffraction is a form of wave diffraction that occurs when field waves are passed through an aperture or slit. In this experiment you will study quantitatively and qualitatively various diffracting objects and their diffraction patterns, by using a laser as a source of monochromatic light and a series of apertures, aligned on an optical bench. Experimental skills • Using a HeNe laser, and taking relevant safety considerations. • Careful experimental alignment and set-up using an optical bench. • Making use of observations and trial/survey experiments (as mentioned in Experiment 3) prior to

taking detailed measurements. Wider Applications • Any real optical system (a microscope, a telescope, a camera) contains finite sized components

and apertures. These give rise to diffraction effects and fundamentally limit the obtainable resolution of any optical device. (There may be other optical imperfections too, such as scratches or misalignment.)

• Thus, the resolution of a given instrument is proportional to the size of its objective, and inversely proportional to the wavelength of the light being observed.

• An optical system with the ability to produce images with angular resolution as good as the instrument's theoretical limit is said to be diffraction limited. In astronomy, a diffraction-limited observation is achievable with space-based telescopes, of suitable size.

1. Introduction Diffraction is the name given to the modification of a wavefront as it passes through some region in which there is a diffracting object. The object is usually an obstacle or an aperture in an opaque sheet of material. Huygens’ Principle postulates that all points on the modified wavefront act as secondary sources of radiation. According to Figure 1, at any point P beyond the object the secondary waves superpose, or interfere, to give a resulting disturbance which is characteristic of the diffracting object. This resulting disturbance is usually referred to as the diffraction pattern of the object, although interference pattern would be a better name.

36

Figure 1: Diffraction through a slit

The form of the diffraction pattern also depends on the distance, D, of the observation plane from the object. Diffraction effects can be divided conveniently into two categories. (1) Near-field, or Fresnel diffraction, for which D is fairly small

(2) Far-field, or Fraunhofer diffraction, for which D >>a 2λ , where a is the size of diffracting unit

and λ is the wavelength of the scattered radiation. In this experiment you will be concerned only with Fraunhofer diffraction effects. The experiment consists of studying, either quantitatively or qualitatively or both, various diffracting objects and their diffraction patterns. 2. Experimental set-up and adjustment of the apparatus 2.1 The laser The source of radiation is a 1 mW helium-neon (HeNe) laser which emits a coherent beam of light of approximately 4 mm2 cross-sectional area. Switch on the laser and adjust it so that the beam is travelling parallel to the longitudinal axis of the optical bench. Make a crude adjustment first by standing back and using your eye to judge how parallel the the axis of the laser is to the optical bench. Then, fine adjustment can be made by checking the beam position on a piece of white card as it is moved along the optical bench. Hold the white card in one of the holders provided and check that the beam strikes the card at the same point, which may be marked with a cross, wherever along the bench it is. Make adjustments using the vertical and transverse fine adjustment knobs on the laser baseplate. Don’t spend too much time doing this; if you’re having trouble, talk to a demonstrator.

37

2.2 Objects and holder Mount the three-jaw slide holder in a saddle positioned close to the laser. You are provided with a series of mounted 2” x 2” slides, etched into which are various diffracting objects. These slides are unprotected and must only be handled by their edges to avoid damage. Diffacting object(s) SLIDE 1 One-dimensional diffraction grating. SLIDE 2 Double slits SLIDE 3 A series of single slits of different widths. SLIDE 4 Two-dimensional diffraction grating. SLIDE 5 One-dimensional diffraction grating 3. Measurement of the width of the central peak Place slide 3 in the slide holder and mount it close to the laser at one end of the bench. Adjust it horizontally until the light is passing through slit C and displaying a clear diffraction pattern on the wall. Always look along the bench, away from the laser when making adjustments. Measure the distance, D, between the slide and the wall. Observe the pattern on the wall and sketch it, to scale, in your lab book. Is the pattern what you expect? What is the diffracting object? Accurately measure the width of the central peak, W. The peak width W is given by: W = Kan, [1] where K depends on D and λ , and a is the width of the slit (Figure 6.1). Repeat this measurement for slits D, E, F and G. Compare the width of the central peak with the slit widths, which are given inµm , on the packet containing the slides. (Record all measurements in metres!) Rearrange equation [1] so that a plot of W as a function of a will give you a straight line graph and, using appropriate graph paper, plot a graph to find the integer n. What do you think is the relationship between K, D and λ ? (Hint: use dimensional analysis to work it out and then refer to the literature to check the correct equation.) 4. Determination of the wavelength of the laser light Now use SLIDE 1 to obtain the diffraction pattern as illustrated in Figure 2. Using the travelling microscope and the Rayleigh mean method (if in doubt, ask a demonstrator), determine the repeat distance d of this one-dimensional grating. Place the slide in the slide holder so that the grating is illuminated by the laser and the diffracted beams lie approximately in a horizontal plane. Maximise the size of this pattern so that you can easily determine the zeroth order (centre) and as many higher orders as possible. Sketch and describe the pattern.

38

Now, by careful experimental measurement it should be possible to determine the wavelength of the laser light. The wavelength λ of the light from the laser is given by

md mθ

λsin

= , [2]

where the angle ?m is indicated in Figure 2

Figure 2: Defining d and ?m

Because ?m is small, Dmx

mm)(

tansin == θθ , and [2] becomes

mmx

Dd )(

=λ [3]

Note x(m) is the distance between the centre of the pattern and the mth diffraction spot. Rearrange the equation to plot a suitable straight line graph in order to determine λ , the wavelength of the HeNe laser. Check that your answer is sensible! 5. Two dimensional grating

SLIDE 4, is a two-dimensional diffraction grating. Use any convenient diffraction method to find the ratio of the repeat distances in the two principal directions. Remember to sketch your observations and discuss.

39

Experiment 18: Measurement of e/m. Introduction This experiment, devised by J.J. Thompson in 1897, allows the ratio of the charge, e, of an electron to its mass, m, to be measured using a cathode ray tube. This is done by producing a beam of electrons (so-called cathode rays) in the form of a narrow ribbon from an electron gun in an evacuated glass bulb. The electron beam is intercepted by a flat mica sheet, one side of which is coated with a luminescent screen and the other side is printed with a centimetre graticule. By this means the path followed by the electrons is made visible. There are two basic methods by which e/m may be determined with the cathode ray tube. They are both based on the equations describing the forces exerted by electric and magnetic fields on moving charged particles. You will try both methods. In both methods, the beam of electrons emitted by the filament passes from right to left to strike the mica screen. We need an expression for the speed, v, of the electrons in terms of the accelerating voltage, Va, between the filament and anode. If the electrons are emitted from the filament with zero kinetic energy and move in a good vacuum, their kinetic energy is just given by

aeVmv

=2

2

(1)

so that v can be found. We shall use this expression later.

Figure 1: Schematic diagram of apparatus

40

Take care! High voltages and delicate evacuated glassware are used in these experiments. PLEASE READ YOUR "CODE OF PRACTICE FOR TEACHlNG LABORATORIES" SHEET Method I - Electrostatic and Magnetic Deflection In this method, the lower deflector plate is connected to the point marked I in Figure 1. A magnetic field B is applied with "Helmholtz coils" (described below). If this magnetic field points out of the plane of the diagram, there will be a downward force on the electrons (use Fleming's left-hand rule) equal to BevFmagnetic =

where e is electron charge and v is their speed. At the same time, by connecting the plates so as to put a voltage VP across them (see diagram) an upward electrostatic force can be applied to the electrons, equal to

de

VEeF Pelectric ==

where E is the electric field between the plates and d is their distance apart. In this experiment, E and B are adjusted so that there is no net deflection of the electron beam, so that the magnetic and electric forces must balance:

Bevde

VP =

and this gives, with equation (1), an expression for e/m

22

2

2 dVBV

me

a

P=

In fact, with the connections as shown, because the lower deflector plate is connected to the cathode while the upper plate is connected to the anode, the plate voltage is equal to the accelerating voltage, VP = Va, so that the previous equation simplifies to

222 dB

Vme a=

41

Procedure For a range of anode voltages, adjust the current through the Helmholtz coils to reduce the electron deflection to zero. The magnetic field in each case is calculated as described below. Tabulate your values of Va, I and B. Plot Va against B2 and hence determine e/m. Estimate the precision of all your measurements and results. What do you think are the main sources of error? Your graph should, of course, be a straight line passing through the origin . Comment on any deviation from this. Method II - Magnetic Deflection onlv In this method, the lower deflector plate is connected to the point marked II in Figure 1 This means that the deflector plates are effectively not used in this experiment. If no compensating electric field is applied, the electron beam will be deflected into a circular path of radius r. Equating the magnetic force causing the deflection to the centripetal force gives

r

mvBev

2

=

Combining this with equation (1) therefore gives

22

2rB

Vme a=

The advantage of this method is that it does not depend on deflection plates. It is very difficult to make deflection plates which have a sufficiently uniform electric field between them, and this leads to a systematic error in the determination of e/m. The only disadvantage of using this formula is that the value of r must be measured. To do this you can use the following relation for circles passing through the origin (which is at the exit aperture of the anode) and the points (x, ±y) on the graticule:

( )

yyx

r2

22 +=

(Note: The origin of the graticule in some tubes is not exactly at the anode and a correction should therefore be made). Derive the above equation. Procedure

42

As in the first method, choose several values of anode voltage. It is then easiest to adjust the current through the Helmholtz coils to produce a particular, easily measurable, radius of the electron beam path. For example, you could make the beam always pass through the point (10.0, ±2.6) cm. The magnetic field is calculated as described below. Note down the values of x, ± y and r and tabulate your values of Va, ± I, ± B. Estimate the precision of all your measurements and results. Plot Va against B2. Choose another value of r and repeat. Repeat for further positive and negative values of r (To get both positive and negative deflections, you will need to reverse the Helmholtz coil current). Calculate e/m for each r and compare and comment on your results. Helmholtz Coils The magnetic field acting on the electrons is provided by a so-called Helmholtz pair of coils each of a radius R, with their centres separated by a distance equal to their radius R. Such a configuration gives a substantially uniform magnetic field in the central region of the coils. The magnetic field B can be calculated from the formula

RNI

B 023

54 µ

= or

RNI

B 0716.0 µ≈

where µ0 = 1.26 x 10-6 TmA-1 (or 4π x 10-7 henry metre-1) N = number of turns on each coil (320 turns of 22 swg enamelled copper wire in this case). I = current through the coils in ampere. The mean coil diameter is 13.6 cm in this case, so R = 0.068 m. The start of each coil is connected to the 4mm socket (A) on the side of the coil bobbin, and the finish to the 4mm socket (Z). For this experiment, in order that the field of the coils should add, connect the power supply to sockets A, with sockets Z interconnected. DO NOT EXCEED A COIL CURRENT OF l.5A FOR MORE THAN 10 MINUTES. FOR LONGER PERIODS OF TIME, DO NOT EXCEED 1.0A.

43

Experiment 19: X-rays 1. Introduction Safety Aspects: Intense X-ray beams are harmful to human tissue. The protective cover of the equipment is interlocked such that the X-ray beam is switched off when the cover is opened. THE CRYSTALS ARE FRAGILE - TREAT THEM WITH CARE. X-rays are electromagnetic radiation of shorter wavelength than light. X-rays have wavelengths of about 0.1 nm whereas light has wavelengths of about 500 nm. X-radiation has many important uses, for example study of the structure of solids. The use of X-rays for this purpose will be explored in Semester 2. However, in this experiment some of the properties of X-rays and their interaction with matter are studied. The experiment consists of the following parts. (a) Measurement of the X-radiation spectrum emitted by a copper-target X-ray tube. (b) Study of the effect of passage through the foils of various elements on the emission spectrum of

the X-ray tube. (c) Measurement of the absorption characteristics of various elements. The wavelength of electromagnetic radiation is usually measured by means of a diffraction grating. In order to obtain reasonable angles of diffraction the spacing of the grating elements must be of the same order of size as the wavelength of the radiation. The spacing of the atoms in simple crystals are typically of the order of 0.1 nm and the electrons in the atoms scatter the X-rays. Consequently simple crystals make convenient diffraction gratings for X-rays. The grating in this experiment is a crystal of sodium chloride cut in the form of a plate. The atomic-scale structure of sodium chloride consists of alternate sodium and chlorine ions arranged in a face-centred cubic arrangement. The arrangement of ions and its orientation with respect to the plane of the crystal plate used here is shown in Figure 1, where a is the basic cube repeat distance.

Figure 1: Arrangement of ions in plate of sodium chloride

44

The relation between the spacing of the ions for the above arrangement, the wavelength λ of the X-rays and the angle 2θ of diffraction is θλ sin2am = (1) where m is the order of diffraction. For the face-centred-cubic arrangement of sodium chloride the lowest non-zero-intensity order of diffraction (other than m = 0) is m = 2 and only even orders are possible. 2. The X-ray apparatus .This consists mainly of a copper-anode X-ray tube, specimen holder and detector carriage enclosed in an X-ray proof housing (Figure 2).

Figure 2: The X-ray apparatus

The white indicator light indicates power on (mainly the supply to the X-ray tube filament); the red indicator, and the audible warning, indicate high voltage and X-rays on. The X-rays can be produced only when the X-ray-proof cover is closed. The whole apparatus is interlocked and is entirely safe. Its use will be outlined by the demonstrator. 3. Measurement of the X-ray tube emission spectrum The wavelength of the X-rays is measured by measuring the angle 2θ at which they are diffracted in the second order (the lowest non-zero-intensity order) from the sodium chloride crystal. For sodium chloride a = 0.564 nm, and for m = 2, so show that equation (1) becomes θλ sin564.0= (nm) (2)

45

Clamp the sodium chloride crystal in the specimen holder (Figure 2) with its long axis vertical and with the largest ground face of the crystal in the X-ray beam. Insert the 1 mm slit primary-beam collimator into the X-ray tube housing with the slit vertical and place the 1 mm slit diffracted-beam collimator in the detector carriage position 18. View the crystal face, and the primary-beam collimator slot, through the slit in the diffracted-beam collimator. If necessary, rotate the primary-beam collimator until its slit is parallel to that of the diffracted-beam collimator and to the crystal face. Place the circular-aperture slide in detector carriage position 17 and the Geiger-Muller detector in its holder in position 26 (Figure 3)

Figure 3: Plan view of component arrangement

Set the scaler unit high voltage to 400 V. Move the detector carriage from 25° to 40° in steps of 1° between 25-28°, 1/6° between 28-33° and 1° between 33-40°. The 1/6° steps can be made with the thumb wheel. At each position count for 10s and record the count and angular position 2θ (Figure 4). Use Poisson statistics to estimate the random count error and remember to subtract the systematic error due to the background radiation count by making counts away from a peak or with no source on.

46

Figure 4: Plan view of arrangement during measurement of count rate

47

Plot the count against 2θ to give the spectrum. It will be seen that this consists of a general background of radiation with two prominent peaks. The longer-wavelength peak is called the Kα line, the shorter is the Kβ line. Calculate their wavelengths. What do the peaks represent? What are the widths of the peaks? What might cause this width? 4. Characteristic X-ray spectra X-ray spectra can be considered to arise from transitions between energy levels characterized by a quantum number nl and levels with quantum number n2. The energy associated with each level is given by

2

2)(nZRhc

Enσ−

=

where: R is a universal constant 1.0968 x l07 m-1; Z is the atomic number, which for Cu = 29; h = 6.626 x 10-34 Js; σ is the screening constant. This relationship is identical to that which is applied to the hydrogen spectrum apart from the appearance of the screening constant. This arises because, unlike the case of hydrogen where there is only one electron, a given electron experiences a field due to the charge on the nucleus modified by the field due to the other electrons. The energy emitted when the atom changes from a state defined by n1 to that defined by n2 is observed as a quantum of frequency υ such that

−−==∆ 2

122

2 11)(

nnZRhchE σν

The Kα line results from a change of n from 2 to 1; the Kβ line results from n = 3 to 1. Use your wavelength data to do the following. (a) Calculate values for σ for the Cu transitions. Comment on any differences you observe. (b) Draw an energy-level diagram for copper (lines separated by a distance proportional to the energy separation and clearly labelled). Use σ calculated for Kβ . What energy would be needed to remove an innermost electron entirely from a copper atom? The K series of lines are so-called because the final state is the n=1 or K shell of the atom. Other shells exist for n = 2, 3 etc and are called the L,M, etc shells. The L series of lines results from transitions which finish at n = 2, so that the Lα line is produced when n = 3 → n = 2. What wavelength would you predict for this line? (Ignore the screening effect). Could you detect such radiation with your apparatus? [Consider equations 1 and 2].

48

5. Effect of Passage through Foils When X-radiation passes through a foil its intensity is reduced according to the equation )exp(0 xII x µ−= where: µ is the linear absorption coefficient; x is the thickness of the foil; I0 is the incident intensity; Ix is the transmitted intensity. Replace the circular aperture slide in the detector carriage position 17 by a copper foil and determine the magnitude of µ for absorption of (a) Kα and (b) Kβ radiation. 6. X-ray Absorption edqes The absorption coefficient is heavily dependent on the wavelength of the X-rays. This can be best understood if we consider the physics of the absorption process. As X-rays pass through matter they interact with the atoms and lose energy. The main energy-loss process is ionization. The X-rays interact with the electrons which are bound to atoms of the absorber and lose the energy which is required to remove the electrons from the atom. This process is described by ..' EBhh −= νν where υ' is the frequency of the X-rays after absorption; υ is the incident frequency and B.E. is the “binding” energy required to ionize the electron concerned. Thus the energy lost in any one such ionization process will depend on: a) the absorbing atomic species; b) the shell from which the electron is removed as the B.E. for an electron n, say, the L shell will differ from that for an electron in the K shell. Consider a simple case where X-rays pass through an absorber which is composed of atoms which have electrons only in the K and L shells. X-rays of the smallest energies (lowest frequency, largest wavelength) will be heavily absorbed by ionizing electrons in the L shell, but will not have enough energy to ionize electrons in the K shell. This process is described by LEhh −= νν ' where EL is the B.E. of the electron in L shell. If the energy of the X-rays is increased, the X-rays become more penetrating and the magnitude of the absorption coefficient falls rapidly. In general, µ ∝ λ3, so that the variation is as shown in figure 5.

49

Figure 5: Variation of absorption coefficient with wavelength and frequency.

At a certain critical frequency (and equivalent wavelength), the X-ray energy has been increased to such an extent that the X-rays are now able to ionize not only electrons in the L shell, but also electrons in the K shell. Thus the absortion coefficient increases very rapidly as shown in figure 6.

Figure 6: Variation of absorption coefficient with wavelength and frequency

for two transitions This discontinuity is termed a K absorption edge and will occur at well defined wavelengths which are characteristic of the absorber concerned. To a good approximation the frequency υK associated with the K absorption edge is given by KK Eh =ν , where EK is the binding energy of the electron in the K shell. Similar absorption edges may occur for L,M, etc shells. Where on a wavelength scale would you expect to find the copper K absorption edge in relationship to the α and β lines? Is this supported by your values for the linear absorption coefficients? Determine the position of the copper absorption edge. Insert the copper foil in position 17 and investigate the intensity of transmitted radiation over a wide range of angles. Plot log10 (I0/Ix ) against 2θ and hence determine the position and wavelength of the edge. How does this compare with your estimate in 4(b)?

50

Experiment 20: Microwaves Safety Although the microwave power used in this experiment is very low students should take care not to look directly into the source when it is switched on. The resistor mounted on the back of the transmitter does get hot after extended use. Outline The properties of waves in general and electromagnetic waves in particular are examined by using microwaves of wavelength ~2.8 cm. The properties examined include polarization, diffraction and interference. The interference experiments are similar to those performed with visible light at much shorter wavelengths (and sound with similar wavelengths). However, the macroscopic wavelength of microwaves is exploited to reveal behaviour not readily accessible at short wavelengths, in particular phase changes on reflection and edge diffraction effects. Experimental skills • Experience of handling microwave radiation, sources and detectors. • Experience of polarized electromagnetic radiation. Wider Applications • Microwave radiation is used in communications, astronomy, radar and cooking.

Mobile phones use two frequency bands at ~ 950 MHz and ~18850 MHz. Astronomy - the cosmic microwave background radiation peaks at ?= 1.9 mm. Microwave ovens use a frequency of 2.45 GHz wavelength of 12.2 cm. The oscillating electric field interacts with the electric dipole in water molecules so that they rotate, have more energy and so get “hotter”. Since water molecules in solid form cannot rotate ice is an inefficient absorber of microwave radiation.

• The manipulation of polarization is an important way to exploit electromagnetic radiation. This is not restricted to plane polarization. For example “circularly” polarized light is exploited in the latest 3D films shown at cinemas.

• Electromagnetic radiation detection is common to many branches of physics. For example with an array of detectors similar to the ones used here and some optics astronomical imaging becomes possible – this is a very active research area within this School.

Equipment List: Microwave generator, two detectors (point probe and horn), Multi-meter (using mV or V scale, depending on equipment), metal plates and grid.

1. Introduction The name “microwave” is generally given to that part of the electromagnetic spectrum with wavelengths in the approximate range 1mm - 100 cm (10-3-1 m). This compares with the visible region with wavelengths of 4 to 8 x 10-7 m. Microwaves therefore have a wavelength which is >20,000 times longer than light waves. Because of this difference it is easier in many cases to demonstrate the wave properties of electromagnetic radiation using microwaves.

51

1.1 Electromagnetic Waves An electromagnetic wave is a transverse variation of electric and magnetic fields as shown in figure 1 and travels through space with the velocity of light (3 x 108 m s-1). Because it is a transverse wave it can be “polarized”, meaning that there is a definite orientation for their oscillations. As shown in Figure 1 an electromagnetic wave is composed of electric and magnetic fields oscillating at right angles. The direction of polarization is defined to be the direction in which the electric field is vibrating. (This is an arbitrary matter; the magnetic field could equally well have been chosen to define the direction of polarization). Plane polarized radiation means that the electric field (or the magnetic field) oscillates in one direction only.

Figure 1 The electric and magnetic fields in an electromagnetic wave. E is the electric field

strength, B the magnetic flux density. The wave propagates with a velocity of 3 x 108 m s-1.

The microwave transmitter provided emits monochromatic plane polarized radiation. A normal light source is a mixture of many different directions of polarization so that its average polarization is zero. An electric field is defined in terms of both an amplitude and direction and is therefore a vector. It is useful to think of polarized radiation in terms of vectors. The detectors of (microwave) electromagnetic radiation used in this experiment are polarization sensitive (some are not). In this case the relative orientation of the transmitter (and electric field) and the detector (receiver) is important and is illustrated in Figure 2.

?

Electric field direction of polarisedelectromagneticradiation

orientation of polarisation sensitive detector

Figure 2. Plane polarised radiation incident at an angle ? with respect to the sensitive direction of the detector.

In Figure 2, if the amplitude of the electric field of the incident radiation is E0 the component that is experienced by the detector is E0cos?. Some detectors give an output that is proportional to the

52

amplitude of electric field, however many have an output proportional to the intensity, I (or power). Intensity is proportional to the square of the electric field, so for an aligned field and detector

I = I0 = kEo2

whereas at an angle, ?,

I = kEo2cos2? = Iocos2?.

From the above, the angular dependence of the signal is capable of revealing something about how the detector/receiver used operates. “Diffraction” and “interference” both relate to the superposition of waves and are essentially the same physical effect. Custom and practice dictates which term is used in a particular circumstance. The essential principles should be familiar to 1st year physics students and will not be repeated here.

2. Experimental

2.1 Apparatus: The Microwave Equipment • The transmitter incorporating a Gunn diode in a waveguide and a horn gives plane polarized*

radiation and is operated at 10 V, fed by a power supply. • There are two receivers*, one is a feed horn receiver the other is a probe. • The feed horn receiver is the most sensitive and is both polarization dependent* and directional. • The probe is non-directional, but is still polarization dependent and is less sensitive. • The receivers are connected to a voltmeter on its mV range.

*The polarization of the transmitter and horn receiver is vertical if the writing on the back of the units is horizontal. The probe receiver placed supported by its stand on the bench is sensitive to vertically polarized radiation.

Important: • Reminder: Do not look into the transmitter when it is turned on. • Neither receiver should be placed nearer than 10 cm from the transmitter. • Stray reflections are a big problem when undertaking microwave experiments. To

minimise these, the experiment should be carried out on the top level of the bench and all objects (bags, hands and arms etc) should be kept out of the beam whilst taking measurements.

2.2. Standing waves and the determination of wavelength To create a stationary (standing) wave a reflecting surface is placed in the path of a progressive wave to reflect the wave along its own path. The resulting waveform should be similar to that shown in Figure 3 where the distance between successive nodes (or antinodes) is half a wavelength.

53

Node

Antinode

Incident wave velocity cReflected wave velocity c

E_/2

Figure 3. Depiction of the standing waves set up when a wave is reflected off a surface.

• The (aluminium) reflector plate should be approximately 1 metre from the microwave source. • Place the probe in the region of the standing waves and move the reflector plate either towards

or away from the transmitter. (A very similar experiment can be performed by moving the detector with the reflector plate fixed.)

• The probe will pass through the wave form given in Figure 3 and when the probe is connected to the meter in the receiver it will display successive maxima and minima.

• Determine the wavelength of the microwave, by recording the position of several maxima and plotting a graph of distance versus maxima (the slope will give a value for half a wavelength). Does the wavelength agree with the value written on the back of the transmitter horn?

2.3 Plane polarised electromagnetic radiation This section consists of a number of experiments to reveal the behaviour of the microwave source and receivers/detectors as well as some of the properties of plane polarized radiation. Plane polarization and receiver sensitivities • Position the transmitter and horn receiver 0.5 m apart with both oriented for vertically polarized

radiation. Align the transmitter and detector by maximising the signal and make a note of the signal.

• The polarization of the emitted radiation and polarized sensitivity of the receiver can be demonstrated by rotating the transmitter through 90o. Find the minimum possible signal and record it.

• Repeat for the probe receiver and compare the properties of the two receivers. • Return the transmitter and horn to their vertical position. Place the large metal grid between the

two, rotate it and observe the variation in the received signal. What effect does the grid have? Why?

Detection of polarized radiation: angular dependence Either by using the metal grid or by rotation of the transmitter, deduce the dependence of the measured power on the angle of polarization. (This may be quite tricky.) • Find a suitable way of measuring the angle of rotation and vary this in 15 degree steps from 0 o

to 180 o. Record the measured signal. • Tabulate the signal measurements along with the expected values for cos? and cos2?

dependencies. What do the results imply?

2.4 Demonstration of interference effects

54

This part of the experiment builds up a microwave analogue of the single slit optical interference experiments. By concentrating on the straight through beam the experiment complements optical diffraction experiments. The general arrangement is shown in Figure 4.

transmitter

x

A

A'

to meter onreceiver

Figure 4. Schematic of the experimental arrangement for interference from a single slit (the transmitter is shown relatively much closer to the slit than is required) The experiment is performed in four parts whilst keeping the distance between the front of the transmitter and the plane AA’ constant (at ~0.6 m). This will allow all results to be compared.

(i) No slits in place This section gives an indication of the spread of microwaves emitted from the source.

• Position a 1 m rule on the bench top to provide an indication of position in the AA’ plane. • Moving the probe in 2 cm steps between measurements, take 8 measurements either side of the

centre line, i.e. 17 measurements in all. • Plot the data. Note: The graph shows the distribution of microwave power in the “beam”

emitted from the transmitter.

(ii) Single slit: variable slit width probe fixed in straight through position This section investigates the effect of slit width on the straight through beam.

• Position the two large plates equidistant from the front of the transmitter and the plane AA’, with a slit width of 3 cm.

• Keeping the centre of the slit on the line between transmitter and probe, take measurements as the separation of the plates (width of the slit) is increased in 2 cm steps up to ~21 cm and then in 1 cm steps up to ~35 cm.

• Plot the data and compare with (i).

Note: The above results have all the hallmarks of interference.

(iii) Single plate: variable plate position, probe fixed in straight through position This section seeks to provide an explanation for the results found in (ii).

55

• Position one large plate as above but with one of its edges directly in the line of sight between the source and the detector. Make a note of this position and then move it across a further 5 cm to obscure the detector.

• From this starting position take readings as the probe is moved out of the beam. Take readings every ~2 cm for the first 10 cm and every 1 cm for the final 10 cm (20 cm movement in total). (You can always add more readings if you need to.)

• Plot the data and consider whether two such single plates can explain the results in (ii).

Note: There is very little scattering of radiation behind the plate.

The origin of interference If all has gone well, the two plate/single slit the interference behaviour of the straight through beam can now be understood to arise from the addition of the effect of two single plates. The single plate behaviour is better considered to be an example of “straight edge diffraction” where the straight through beam from the emitter interferes with a secondary source of radiation reflected from the edge of the plate. As the plate is moved away from the centre line the path difference, between the straight through and reflected beams, increases. From this argument it might be expected that the first turning point, corresponding to a path difference of ?/2 (phase difference of p), would be a minimum, whereas clearly it is a maximum. This is explained by the reflection at the edge producing a (negative) phase shift in the re-emitted radiation.

• If you have time, use Pythagoras theorem to determine the phase shift** caused by reflection at the edge. See Appendix at end.

** A simple reflection (as in 2.2) would be expected to result in a -p phase shift, however with this geometry the Gouy effect is reported to result in a further -p/4 phase shift giving a total of -3p/4.

(iv) Single slit diffraction pattern: fixed width This section seeks to illustrate the fundamental equivalence of light and microwaves by generating a (familiar) single slit diffraction pattern.

• Position the two large plates as in (ii) but with a separation of 11 cm. • Moving the probe in 2 cm steps between measurements, take 8 measurements either side of the

centre line, i.e. 17 measurements in all. • Plot the data and compare the first minimum with its expected position (given ? = 2.8 cm). (Note: Here due to diffraction, minima are expected at nλ = d.sinθ, where d is the slit width.)

Appendix The experimental arrangement is shown in figure 6 where the source is considered to be a point - a parallel beam would be more appropriate for a visible laser/edge arrangement. The distance from plane of sheet to the source and detector is the same.

56

Detector

Metal sheet

Source

L

d

d

Figure 6. Schematic of experimental arrangement for edge interference. The paths for microwaves travelling directly between source and detector and via the edge are shown. The geometric path difference (found using Pythagoras) is 2d where

LLd −+= 2122 )(δ Extrema (i.e. maxima and minima) in intensity occur, taking into account the Gouy effect when:

8/32)(22/)1( 2122 λλ −−+=− LLdm where m is a positive integer. Note half wavelength path lengths give alternating max and min and so the “extrema”.

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Experiment 21: Computer Error Simulations and Analysis Outline The autumn semester introduced random errors (from repeated measurement and from straight line graphs) and the propagation of errors (through techniques of partial differentials and adding in “quadrature”). Having used these concepts for a while, this session revisits the underlying concepts using new and existing Python computing skills.

Experimental (and computing) skills • Understanding the statistical analysis of data. • Use of statistical computing tools.

Wider Applications This experiment illustrates the unseen statistics behind all practical physics • In advanced applications the statistical analysis of data is all handled by computers. • This section explores the nature of least squares fitting and provides an introduction to

alternative numerical approaches.

1. Introduction The experiment “Statistics of experimental data (Gaussian Distribution)” performed during the autumn semester (PX1123) introduced you to some of the underlying foundations of the analysis of random errors. Here the subject is revisited. But, by making use of a computer (and Python programming), to both generate and analyse data much faster progress can be made. After reconsidering the error associated with repeated measurements of a single point, the session moves on to consider the treatment of error propagation (the combination of errors) and the “least squares” analysis of straight line data.

Session 1. Evolution of errors with repeated measurement with a normal distribution. 2. Error propagation (making sense of adding in quadrature) 3. The statistics of straight line graphs

Quick Reminder: the nature of experimental measurements (see section III.2 of PX1123 lab manual for full treatments) • Repeated measurements usually result in a normal distribution around a mean value. • With a reasonably large number of repeats “standard errors” represent the uncertainty in

determined values. • For y(x) when x is varied the data points can be considered as very similar to repeats with the

points distributed above and below the “best fit line”.

2. Experiments It will be a good idea to have access to the website during the course of the session. This should be one of your “favourites” but if it is not:

https://alexandria.astro.cf.ac.uk/Joomla-python/ Quick Python reminder – relevant syntax is present in week 2 and 3 (Arrays, Vector Algebra and Graph Plotting) of the taught computing course.

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2.1. Normal/Gaussian statistics of repeated measurements Section 2.1 will be based on the simulation of repeated measurements of two timed events, A and B both measured with a stopwatch. Suppose that: • For the sake of the simulations the true values of A and B are 2.0 s and 3.0 s exactly. • The standard deviation* that characterises both measurements is 0.2 s.

*The standard deviation parameterises the spread in values that are obtained and so is also said to characterise (parameterise) the precision of the measurement.

2.1.1 Distributions for A and B The first step is to create arrays of points for A and B randomly generated from ideal normal distributions. The first point in each array then corresponds to the first measurement etc. Provided these arrays are only created once the subsequent analysis can be cross compared. To achieve this arrays for A and B will be created in the Spyder console. This does not exclude creating programmes in the editor because they can (and are normally) executed in the console and so can call on arrays that exist there.

Creating arrays This will be done using the normal() function. As given in the object explorer the defaults for this are:

normal(loc=0,scale = 1.0,size =1 value)

where loc is the mean value of the distribution, scale is the standard deviation and size is the number of points.

Do the following: • Create n = 1000 point arrays for A and B (labelled as A and B) • Create and print out a single (20 bin is appropriate) histogram including both A and B and

comment on the range of values for each and any overlap between the distributions. • Perform a statistical analysis of A to find the mean, standard deviation and standard error. • Transfer these to the editor and save the code as a (very) simple programme – it is worth it as it

will be used a few times today. Since this runs in the “Console” it can call on the A array generated earlier. Do not write a function to generate A in the programme as this will overwrite it.

• Change the array name in the programme to analyse the B array. • Consider the appropriate parameter to use as the errors in A and B, state their values (with

errors – as usual) and state whether they agree with the accepted/known values of A and B.

2.1.2 Error propagation (adding in quadrature) Students have been required to combine errors based on the outcomes of partial differentiation (which hopefully makes sense) and addition in quadrature (which hasn’t yet been justified). The aim here is to justify the addition in quadrature. The addition and multiplication of two values (A and B) will be considered and their errors will be taken to be their standard deviations. (A large number of points (n) will be used so standard errors are more appropriate however since the two are linked by a factor of (n-1)0.5 this will not affect the interpretation or error propagation).

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Addition of A and B (Sum, S=A+B) Reminder: error propagation for P = A + B

Partial differentiation gives and

Or and Combining the (or ?P) contributions in quadrature gives the familiar

Here a distribution of n (=1000) measurements of S = A + B will be generated, i.e. the first value of S is the first measurement of A is added to the first of B and generally for the ith term Si = Ai +Bi. In this way some errors/deviations from the true value will reinforce positively or negatively and some will tend to cancel. This is as would be expected in a real experiment.

• Add the arrays A and B together to create the S array. • Plot a histogram and perform a statistical analysis of S to find its mean and standard deviation. • Compare the mean of S with the expected value and its standard deviation with the error in S

calculated (in the usual way) using the standard deviations in A and B as their errors.

Multiplication of A and B (product, P = AB) Reminder: error propagation for P = AB

Partial differentiation gives and

Or and Combining the (or ?P) contributions in quadrature

Dividing by P2 = (AB)2 gives the familiar

Here a distribution of n (=1000) measurements of P = AB will be generated, i.e. the first value of P is the first measurement of A is multiplied with the first of B and generally for the ith term Si = Ai.Bi. Again, some errors/deviations from the true value will reinforce positively or negatively and some will tend to cancel. • Use the same arrays for A and B as before. • Multiply the A and B arrays together to produce P. • Plot a histogram and perform a statistical analysis of P to find its mean and standard deviation. • Compare the mean of S with the expected value and its standard deviation with the error in S

calculated in the usual way.

2.1.3 Evolution of mean standard deviation and standard error The aim here is to illustrate the difference between standard deviation and standard error and their suitability in representing the random error in measurements.

The A array of 1000 points generated at the start of this section will again be used and should not be overwritten. The approach will mimic an experiment in which the number of measurements is gradually increased and the mean, standard deviation and standard error evolve.

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The Python programme written earlier needs to be modified to perform the analysis in this section. To do this elegantly requires the use of “For loops” which is scheduled for week 7 (but subject to change). Depending on proficiency (and perhaps confidence) students may use loops (a) or stick to a simpler sampling strategy (b).

For both strategies it will be necessary to sample (or return) parts of the array A, a sequence that always starts with the first value. This skill was addressed in week 3 of the computing course. Start by testing that you can sample the array correctly.

(a) Simple sampling strategy • Transfer the code to sample the array to your existing programme and test that it performs

correctly (eg by examining the mean of a small number of points). • Next run the program to analyse the first 5, 10, 20, 50, 100, 200, 500, 1000 points. • Plot a graph of (mean value – 2), +/- standard deviation and standard error on the y –axis and

number of samples (measurements) on the x-axis. (+/- are plotted here to represent possible error ranges).

• Consider and describe the evolution with number of measurements.

(b) Advanced strategy (using For loops) • By using a For loop it is possible to sample and analyse each measurement from 2 to 1000

points and see the evolution in much finer detail. • However, do not attempt this approach unless you are proficient in the use of loops. • Consider and describe the evolution with number of measurements.

2.2 Straight line graphs Laboratory and computing courses have introduced the analytical method of finding the “least squares” best fit (and associated errors) to straight line (linear) data. Although this has been used it has not yet been examined in detail. To do this the “Hooke’s law data”, given in Table 1, used in the computing module will be used as an example data set.

Mass (x_data)/kg Length (y_data)/m 0 0.055

0.1 0.074 0.2 0.089 0.4 0.124 0.5 0.135 0.6 0.181 0.8 0.193

Table 1: Hooke’s Law data taken from the computing course

Least squares analysis leads to gradient = 0.18+/-0.01 m/kg and y intercept = 0.055 +/- 0.006 m, so that the best estimate of the straight line representing the data is y = 0.18x +0.005.

Reminder of the “least squares” approach. • The errors in x points are insignificant – this means that the deviation of a point from the fit line

can be taken to be solely associated with the y values. Consequently the statistics describing

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this situation are essentially the same as those describing repeated measurements of a single point.

• The (random) errors characterising the y data points are all the same (and can be described by a standard deviation) – this means that all points have equal importance or “weight”.

• The best fit line must pass through the mean of the x and y data values (x_mean and y_mean respectively).

• Since the errors in x points are insignificant the difference between the best fit line and the data points is characterised by the difference between the corresponding y values, known as “residuals”. The values of m and c when the square of the residuals is minimised is the best fit line.

Note: the least squares method of obtaining best fits is not limited to straight line data although it is then more difficult or impossible to find analytical expressions and it is often necessary to resort to numerical techniques (through use of a computer).

The approach for investigating least squares fitting of straight line graphs A set of straight lines all passing through the mean of the x and y data values but having different gradients (including the best fit gradient) will be generated. The square of the residuals will be calculated for each line and plotted against gradient.

Guided be the known best fit we’ll consider the quality of fits for gradients of m = 0.18 +/- 0.05 m/kg, i.e. in the range 0.13 to 0.23 m/kg in 0.01 m/kg steps.

Do the following In the Spyder console: • Generate arrays of x and y data points, call these x_data and y_data. • Find the mean of the measured x and y points. • For m = 0.18 m/kg (we’ll start with the best fit gradient) calculate an array of points for the

corresponding straight line based on the x_data points. • Generate an array of the difference between the y data points and the y best line points. These

values are the residuals. • Square the residuals and find their sum and record this in a table in your diary. • Transfer the working code to the editor to create and save a little programme. • Repeat* the calculation for all the required gradients. • Plot a graph of sums of the squares of residuals versus gradient. • Describe its form. * This could also be done using a loop.

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III: BACKGROUND NOTES III.1: Experimental Notes:

INTRODUCTION TO ELECTRONICS EXPERIMENTS

In these experiments you will be required to build a variety of analogue electrical circuits and to make measurements of potential differences, current flows etc. The following notes give advice on building circuits and how to use test equipment, such as oscilloscopes, multimeters and signal generators. The final section gives advice on eliminating faults in electrical circuits.

1. Building Circuits

BREADBOARDS are used to make circuits in some experiments. This is a purpose-built board which allows you to make all the necessary connections between components by means of plugs and sockets and eliminates the need for soldering. Figure 1 shows a diagram of a breadboard of the type you will use.

Figure 1: The breadboard you will use in Yr 1experiments with details of connections.

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At the top of the breadboard are a set of connections which can be connected by 4mm connectors or by bare wire if the tab highlighted is pushed in. There is a choice of having a variable DC voltage or a constant voltage given by the yellow/green/blue and red/black respectively. The green plug is the ground socket, and the range of voltages offered by the variable power supply is between

11.5V. The grid of blue sockets has its own methodical set up too. Sets of 5 horizontal sockets are connected within themselves, but are independent of the sets above and below. Furthermore sockets within a vertical column are connected, as there are four of these vertical sets, it can be useful to set one to 0V, one to positive voltages and one to negative voltages. As a result, you must think about the points at which you connect a wire, as it needs to be in the appropriate row or column in order to complete the circuit. You are advised to construct circuits so that they resemble as near as possible the circuit diagrams in the script. You will find this of great benefit when trying to locate faults. Note that two interconnecting wires are indicated by a dot placed at their intersection in a circuit diagram. Wires which simply cross each other are not connected. 2. The Oscilloscope The basic functions of the oscilloscope are shown in Figure 2. Most of the functions are self explanatory.

Figure 2. Front Panel of the GwInstek Digital Storage Oscilloscope Basic functionality is controlled by: Function Keys: Accesses the function alongside the button shown on the LCD display Variable Knob: Increases or decreases a value and moves to the next or previous parameter CH1/CH2/Math: Configures the vertical scale and coupling for each channel input (CH1 and CH2), and also Math operations such as ‘add’, ‘subtract’, or perform ‘Fast Fourier Transforms (FFT)’ on input waveforms Volts/Div: Sets the y axis scale Time/Div Knob: Sets the timebase (x-axis scale)

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Autoset Key: Automatically configures the horizontal, vertical, and trigger settings according to the input signal. Trigger Level Knob: Sets the trigger level. This controls the scope's ability to reproduce a steady trace on the screen. Additional Notes on Timebase trigger For the analysis of time varying voltages the trace on the oscilloscope screen must be stationary. If the timebase were "free-running", that is, not synchronised to some multiple of the repeat-time or period of the input waveform then the trace on the screen would not be stable. To synchronise the timebase to the repeat time or period of the input waveform a "trigger" is used. The trigger circuit in the oscilloscope effectively 'fires' or emits a pulse when the input voltage passes a set threshold level. This pulse is then used to initiate the timebase cycle. The input to the trigger circuitry is normally taken from the y axis input amplifier. Sometimes it is found necessary to apply an alternative, externally-derived voltage direct to the trigger circuit via the external trigger input. The trigger is sensitive to both slope and polarity of the input waveform and can be set to fire on a particular slope and on positive or negative polarity. Hence, if a periodic waveform such as a sinusoid is applied to the input terminals, the trigger can be set to fire once every cycle at a fixed point in the cycle (Figure 3). The timebase cycle shown would lead to a stationary trace representing one cycle of the input waveform. The trigger level is shown on the display on the RHS of the axis (small arrow marker). This is the trigger threshold voltage shown in figure 3.

Figure 3: Understanding the timebase

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Notes on the AC and DC components of the oscilloscope waveform.

A general time-varying voltage such as that shown in Figure 4(a) may be divided into two components: (i) a D.C. component, equal in magnitude to the mean value (ie, the average over all time) of the

waveform (Figure 4(b)) and (ii) an A.C. component which remains when the D.C. component has been removed from the

waveform (Figure 4(c)). The oscilloscope amplifiers may be D.C. or A.C. coupled. Try this on the waveform you are observing. When the coupling is set to D.C. the trace represents both the D.C. and A.C. components as shown in Figure 4(a). Setting the coupling to A.C. removes the D.C. component just leaving the A.C. component as in Figure 4(c).

Figure 4(a)

Figure 4(b)

Figure 4(c)

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3. The Multimeter The multimeter you will encounter in your first year experiments (and many subsequest) is a hand held digital device shown in figure 5. It is capable of measuring direct and alternating voltages and currents, resistance, and diode readout. You must select the mode of operation on a central switch, apply your terminals correctly and select the appropriate measuring range.

Figure 5: The Multimeter

4. The Signal Generator The output from the oscillator is available from the bottom right BNC socket. The signal amplitude can be varied by means of the attenuator (O dB or -20 dB) and the variable output level. Three different waveforms are available: sine, triangular and square. The OFFSET knob works only when the DC OFFSET button is depressed. 5. Resistance Colour Codes Resistors are colour-coded to indicate their resistance, tolerance and power-handling capacity. The background colour indicates the maximum power of the device. You will use only 0.5 W resistors (dark red background). The four coloured bands can be read as described below to determine the resistance and tolerance. The final gold or silver band gives the tolerance as follows: gold ± 5% silver ± 10%

Terminals

Rotary Switch

Range Button

Display

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Digit Colour Multiplier No. of zeros silver 0.01 -2 gold 0.1 -1 0 black 1 0 1 brown 10 1 2 red 100 2 3 orange 1 k 3 4 yellow 10 k 4 5 green 100 k 5 6 blue 1 M 6 7 violet 10 M 7 8 grey 9 white

Table 1.1: Resistor colour-codes Example: red-yellow-orange-gold is a 24 kΩ, 5% resistor. 6. Finding Faults in Electronic Circuits During the course of the laboratory work you will probably encounter practical difficulties. You should always try to solve these problems yourself, but if you are unable then you should call on the assistance of the demonstrator. Occasionally, a circuit will fail to operate because of a faulty component, but more often than not problems arise from the incorrect use of test equipment, the omission of power supplies from circuits, or the use of broken test leads. Faults are not usually apparent to the naked eye, but they may be detected quite easily by following a systematic checking procedure such as that outlined below. If after following these procedures your circuit still doesn't work, then DO NOT HESITATE TO ASK THE DEMONSTRATOR FOR HELP. (i) Ensure that you understand how to use each piece of test equipment. If in doubt, consult the

demonstrator. (ii) Examine the circuit for any obvious faults. Is the circuit identical to the circuit diagram in the

script? Are the components the correct values? Are there any loose wires or connectors which could short out part of the circuit?

(iii) The fault may lie in the circuit itself, in the signal generator which supplies the input signal, or in

the measuring equipment. Switch on the power supply to the circuit and apply the input signal. Use both channels of a double-beam scope to measure simultaneously the input and output signals of the circuit. Check at this stage to see whether the scope leads are faulty. Ensuring

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that you do not earth any signals (see next section), connect the scope to the input and output of the test circuit. If there is no input signal, disconnect the signal generator and test it on its own. If the generator functions only when disconnected from the circuit, it implies that the fault lies in the circuit and that it is possibly some type of short circuit, most likely associated with incorrect earthing. If there is an input signal but no output signal, the fault lies in the circuit.

(iv) A common fault which occurs when using more than one piece of mains-powered equipment

is the incorrect connection of earth lines. ALL EARTHS MUST BE CONNECTED TO A COMMON POINT, otherwise the signal may be shorted out.

(v) If you have established that the fault lies in the circuitry, use your scope to examine the

passage of the signal through the circuit. Components which you regard as faulty should be isolated or removed from the circuit for further testing.

(vi) If you trace a fault to a piece of mains-powered equipment, DO NOT ATTEMPT TO

REPAIR THE FAULT YOURSELF. Report the fault to the demonstrator or technician and ask for replacement equipment.

HOW TO USE A VERNIER SCALE

Vernier scales are used on many measuring instruments including the travelling microscope that we will use in the laboratory. We will begin by looking at the general principle of a vernier scale and then look at the particular scale we will use. Figure 5 shows a vernier scale reading zero. Note that the 10 divisions of the vernier have the same length as 9 divisions of the main scale. If the smallest division on the main scale is 1mm then the smallest scale on the vernier must be 0.9mm. This vernier would then have a precision of 0.1mm and results should be quoted to ±0.1mm.

10 0 Main scale

Vernier 0

Figure 5: Vernier Scale Let us see how it works. Examine figure 6. The position of the zero on the vernier scale gives us the reading. Here it is just beyond 2mm so the first part of the reading is 2mm. The second part (to the nearest 0.1mm) is read off at the first point at which the lines on the main scale and the vernier coincide. Here it is the 4th mark on the vernier (don’t count the zero mark). The reading is therefore 2.4 mm.

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0

10 0

Figure 6: using the vernier

To see why examine figure 7, which is an alternative version of figure 6.

x

0

1 0

D1

D2

Figure 7: why a vernier works

In essence we have been finding the distance X, which is simply given by: X = D1 – D2 = 4×1mm - 4×0.9mm = 4 ×0.1mm = 0.4mm So that is the general principle. Let us see how the travelling microscope scale works. In this case the smallest division on the main scale is 1mm, which implies that the smallest division on the vernier is 49/50 mm = 0.02 mm As an example the reading in figure 1.8 is 113.68mm.

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Figure 8: example reading = 113.68mm. Note: unlike the examples in figures 5-7 the vernier is above the main scale.

Best Match

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III.2 USING EXCEL 1. Determining errors from straight line graphs using EXCEL Instructions • Input the data to be analysed into an EXCEL spreadsheet in column form. • Select a 2x2 array of cells anywhere in the spreadsheet (these are the ones highlighted in the

figure below). • In the function/command line type “=linest( ” - presumably “linest” stands for line statistics. • Opening the bracket leads EXCEL to prompt for

known_y’s simply select using mouse, then insert a comma. known+x’s simply select using mouse, then insert a comma. const input 1 (using 0 would force line through the origin) and a comma. stats input 1 (this sets the correct statistics) and close the bracket.

• The command line should look something like: =LINEST(A5:A14,B5:B14,1,1) • To execute the calculation press CTRL,SHIFT and ENTER • Values for m and c and their errors should appear in the selected 2x3 array in the format shown

in the figure below. The “m”, “c” “errors” “R^2” and “reg error” labels have been added for clarity.

• In this case the gradient is m = 2.60 ± 0.04 and the intercept is c = -1.2 ± 1.6, i.e. the straight line passes through the origin within the (standard) error.

• R^2 is the same value as appears on graphs when adding trend lines: it is a correlation coefficient indicating how good a straight line the data represents.

• “Reg Error” is short for “regression error”; it is the standard error of the measured y values compared to the best fit y values. It is analogous to the standard error for repeated measurements of the same value where values are then compared to the mean of the values.

Least squares fitting of straight line data

The data

x x^2 y 0 0 0 m c 1 1 2 2.60301 -1.18577 2 4 11 errors 0.042074 1.647517 3 9 21 0.997914 3.572721 4 16 42 R^2 reg error 5 25 63 6 36 93 7 49 120 8 64 162 9 81 216

Figure Appearance of EXCEL spreadsheet when determining errors in a straight line graph. The selected 2x3 array of cells (in which values were eventually returned) are highlighted.

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2. Making graphs in EXCEL 2007 EXCEL 2007 is substantially different from previous versions and this has caused students (and staff) some problems: there are more options so things are generally a bit more difficult to find. To help, some guidance on basic graphing tasks is given below.

To make a basic graph • Select two or more columns of data either by clicking and dragging or by selecting a column

holding down control and selecting additional columns. The left hand column will be the data for the x-axis no matter what order the data is selected.

• Select “insert” on the toolbar • Select type of graph (usually “scatter”).

To add titles* • With graph selected, in “chart tools” click on “Layout”. • Here click on “axis title”. For the y axis (primary vertical axis title) it is probably best to use

“rotated title”. • You may also want to add a “chart title” (for your diary but not for inclusion in reports!). *You don’t seem to be able to add equations to titles but you can use Word-like formatting: “CTRL =” for subscripts, “CTRL +” for superscripts.

To change the range of data shown • Either select the axis or choose “format axis”. • Or, under “Layout” choose “Axes”, then the axis of interest, then (at the bottom of the list)

“More… axis options”. • Under “axis options” change minimum and/or maximum to fixed (from auto) and select desired

value(s).

Formatting data series (line and marker) • Right click on the required data series on the graph and then choose “format data series” and

choose from the “series options”. • For example to change marker size choose “marker options” set marker type to “built in” then

set “size”. • Alternatively, with the graph selected: under “layout” the required data series can be selected by

use of the drop down box in “current selection” (on the left of the toolbar).

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III.1: CHECKLISTS: DIARY (LAB BOOK) CHECKLIST Date Experiment Title and Number Risk Analysis Brief Introduction Brief description of what you did and how you did it Results (indicating errors in readings) Graphs (where applicable) Error calculations Final statement of results with errors Discussion/Conclusion (including a comparison with accepted results if applicable) FULL ACCOUNT (REPORT) CHECKLIST Date Experiment Title and Number Abstract Introduction Method Results: Use graphs – and don’t forget to describe them. Indication of how errors were determined Final results with errors Discussion Conclusion (including a comparison with accepted results if applicable) Use Appendices if necessary A risk assessment is unnecessary.