Download Year 1 Lab manual (2016-17) - Cardiff Physics and Astronomy

SCHOOL OF PHYSICS AND ASTRONOMY
FIRST YEAR LABORATORY
PX 1223
Introductory Practical Physics II
(Supplemental Lab Book)
Academic Year 2016 - 2017
NAME:
Lab group:
Welcome to the 1st year laboratory, Introductory Practical Physics II, module PX1223 in
the spring semester. You will need to bring this manual with you to every laboratory
session, as it contains the description of the new list of experiments that you undertake
in this semester. It is designed as a supplement to your existing lab manual, and does
not duplicate essential information contained in the manual you received for PX1123.
You are still expected to have pre-read each relevant section prior to coming to your
weekly laboratory session and to have written your Risk Assessment and Aims.
If you cannot find the information that you are looking for, please ask any member of
the teaching team - your Lab Supervisor, the demonstrators or the module organizer (Dr
Phil Buckle, room S/1.24).
Lab Supervisor:
Contact email:
Demonstrators:
1
1
Contents
1
Contents ....................................................................................................................... 2
2
Introduction and logistics of the 1st year laboratory .................................................. 3
2.1
2.1.1
Introduction ................................................................................................... 3
2.1.2
Attendance ..................................................................................................... 3
2.1.3
Submission of coursework. ............................................................................ 3
2.2
Assessment of practical work .............................................................................. 4
2.2.1
The feedback you should expect to receive .................................................. 4
2.2.2
Decile level descriptors .................................................................................. 5
2.2.3
Notes on decile level descriptors ................................................................... 6
2.3
2.3.1
3
Organisation and administration of the laboratory ............................................ 3
Formal reports of experiments ........................................................................... 7
The process of report submission and assessment ....................................... 7
Experiments ................................................................................................................. 8
3.1
Time table and list of experiments ...................................................................... 8
3.2
Checklist............................................................................................................... 9
3.3
Experiment 12: Optical Diffraction ................................................................... 10
3.4
Experiment 13: Propogation of Sound in Gases. .............................................. 14
3.5
Experiment 14: Microwaves ............................................................................. 17
3.6
Experiment 15: Variation of Resistance with Temperature. ............................. 24
3.7
Experiment 16: Resistive and reactive impedances in RC circuits .................... 28
3.8
Experiment 17: Radiation from a Hot Body ...................................................... 37
3.9
Experiment 18: Report writing feedback ......................................................... 40
3.10
Experiment 19: Computer Error Simulations and Analysis .............................. 41
3.11
Experiment 20: Group Easter Challenge - The strength of paper .................... 47
4
Error propagation: A reminder of the general case................................................... 48
5
Lab Diary checklist...................................................................................................... 50
6
Formal report checklist .............................................................................................. 50
2
2
Introduction and logistics of the 1st year laboratory
2.1
2.1.1
Organisation and administration of the laboratory
Introduction
There are 9 laboratory sessions in the spring semester (no lab in week 1 and no formal
lab in week 11). They are designed to continue on from the foundation in the first
semester. ie:1. To provide familiarity and build confidence with a range of apparatus.
2. To provide training in how to perform experiments and teach you the techniques
of scientific measurement.
3. To give you practise in recording your observations and efficiently
communicating your findings to others.
4. To demonstrate theoretical ideas in physics, which you will encounter in your
lecture courses.
5. To understand the important role of experimental physics
2.1.2
Attendance
Class Times. Labs run from 13:30 to 17:30 on Monday, Tuesday and Thursday
afternoons. Students will in general be assigned the same laboratory afternoon as the
autumn semester, however to make numbers more even and therefore comfortable
there will be a limited number of people being moved between sessions.
Attendance at Laboratories. Experimental physics forms an important part of all degree
programmes offered by the School of Physics and Astronomy and is a requirement for
Institute of Physics accreditation. Attendance at all scheduled laboratory classes is
compulsory. Unscheduled absence from laboratories will lead to loss of marks and
possibly failure of the module. It is not always possible to offer summer resits in
laboratory-based modules (see UG Student Handbook Appendix 1).
2.1.3
Submission of coursework.
As with the autumn semester this module is assessed 100% through continual
assessment (in the form of lab diaries and formal reports).
Deadlines are final and late submission will be awarded zero marks without exception.
Coursework is submitted in the same two ways as the first semester, either through the
“post boxes” near the General Office or electronically through Learning Central. Major
pieces of writing (e.g. your formal laboratory report) are submitted electronically to
Turnitin, an electronic system which helps identify plagiarism.
3
2.2
Assessment of practical work
Assessment will follow the same process as the autumn semester
Each experiment and each report will be marked out of 20 in accordance with the
scheme: 16+ = exceptionally good, contains good physicists’ reasoning; 14+ = very good
solid performance; 12+ = good performance which could be improved; 10+ = competent
performance but with some key omissions; 8+ = bare pass; 7- = fail. Your final module
mark (see Undergraduate Handbook) will be made up as follows:
Formal report
Experimental lab diaries
33.3%
66.7%
(Please see PX1123 for more information on Assessment Criterial used in marking).
2.2.1
The feedback you should expect to receive
You will receive feedback on each of your Lab Diary submissions on a weekly basis. This
feedback will be in the form of a single mark out of 20 with additional written notes to
guide you on things you didn’t achieve and improvements you could consider. The
demonstrators will return your work to you personally, thus giving the further
opportunity for verbal feedback and for you to ask questions. At any time, you can ask
the Lab Supervisor for justification of the mark awarded or where you could improve.
Bare in mind that a mark of 14/20 or better is a first-class degree performance whilst
one of less than 8/20 represents a fail. Your markers will base this mark on the Decile
Level Descriptors provided in Table 1. These describe how well the required task must be
performed in order to obtain a certain range of marks. This method is commonly used in
University assessment where there is no model answer and independence is to be
encouraged.
Note that this system considers “lapses” in two types: “major” and “minor” (also
considered in Table 2). It is worth paying attention to these, as marks of 70% or greater
cannot be awarded in the presence of major lapses.
Your individual marks will be recorded on Learning Central for you to review. It is your
responsibility to check that they have been recorded correctly and to contact the
Module Organizer if that is not the case.
Formal reports are marked at the end of semester and a Report and Feedback Sheet will
be given back to you to give a thorough justification for the % mark received.
The expectation is then on YOU to read, understand and use the feedback received in
order to improve your future performance.
4
2.2.2
Decile level descriptors
Table 1. The descriptors and descriptions used in assessing reports and diaries
Decile
Descriptors
Level Descriptions
range
90-100% Outstanding
The assessed work is as good as could reasonably be
expected from a student at this level. It is uniformlyexcellent in meeting the task specifications. It contains
no major lapses and very few (if any) minor lapses.
80-89%
Excellent
Work of very high quality, but not quite as good as could
reasonably be expected from a student at this level. It is
uniformly very good and sometimes excellent in meeting
the task specifications. It contains no major lapses and
few minor lapses.
70-79%
Very good
Taken as a whole the work is very good in meeting the
task specification. It contains no major lapses but does
contain a number of minor lapses.
60-69%
Good
Taken as a whole the work is good in meeting the task
specifications. It may contain a small number of major
and minor lapses, or no major lapses but significant minor
lapses.
50-59%
Satisfactory
Satisfactory work taken as a whole. It is likely to show
significant variability in meeting the task specifications. It
is likely to contain a number of major and minor lapses.
40-49%
Pass
Adequate work taken as a whole. It is likely to have
significant deficiencies in meeting the task specifications.
It is likely that the work will reveal substantial gaps in
understanding and have significant major and minor
lapses.
30-39%
Fail
Insufficient relevant content, serious
errors/omissions/lapses.
20-29%
Insufficient
Little relevant content, extensive errors/omissions/lapses.
10-19%
Unsatisfactory Very little relevant content, extensive
errors/omissions/lapses.
0-9%
Poor
Essentially no relevant content, extensive
errors/omissions/lapses.
Some notes on the above are present on the following page.
5
2.2.3
Notes on decile level descriptors
Major Lapses
The level descriptions above indicate that in order to award a mark greater than 70%
(i.e. of 1st class standard) there should be no “major lapses”. Major lapses are therefore
important in determining the mark awarded and are listed in Table 2.
Table 2 Common major lapses in diaries and reports
Diaries
Reports
From task description
From task description
No, or highly inappropriate, risk
Significant deviation from the format and
assessment.
structure explained in the support
available in Learning Central.
Content is illegible (neatness per se is not Report is not electronically generated.
a requirement).
Content cannot be easily followed or
Lapses that might be major depending on
understood.
circumstances
Lacking clarity and succinctness.
Obvious (e.g. numerical) mistakes in
principle result(s).
Lacking in appropriate experimental
Substantial gaps in understanding.
observations.
Lack of appropriate data analysis.
Lack of proper error consideration.
Lack of appropriate error analysis.
Lack of concluding remarks.
It should be appreciated that major lapses are, in general, not a restriction of marks over
and above those defined in the task*. For example, a diary with “content that is
illegible” will not represent a good record of the experiment performed.
The point of including the term in the descriptors is to help authors and markers in
thinking about and checking reports and diaries.
*An exception applied to diaries: “No, or highly inappropriate, risk assessment” is
considered a major lapse since safety is important. However, the presence of a risk
assessment is not worth (up to) 30%.
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2.3
Formal reports of experiments
The formal report for PX1223 is due in at 4pm on the Friday of teaching week 11 of the
spring semester.
2.3.1

The process of report submission and assessment
Reports are submitted via Turnitin whose plagiarism* checking is later supplemented
by that of the markers.
 Reports are assigned to the 3 lab supervisors to mark.
 Markers read and annotate the scripts, fill in the “Report Mark and Feedback Sheet”
and decide on marks for each section.
 Markers then perform a reality check on the mark; again by comparing their view of
the report against the decile level descriptors, before applying any necessary
adjustments. This check is designed to pick up double awards/penalties that can occur
when using mark sheets (due to the sections not being entirely independent).
 When all reports have been marked the Module Organizer and all Lab Supervisors
meet and moderate the marks: by comparing the averages for different markers and
experiments and second marking a selection of reports.
* Check your student handbook for guidance: although data analysis can be done as
pairs it is advisable to not exchange reports once the writing process begins.
7
3
3.1
Experiments
Time table and list of experiments
Week
Experiment
Title
Spring Semester (PX1223)
1
Free Week
2-8
(see list)
12
13
14
15
16
17
18
Optical Diffraction
Propogation of Sound in Gases.
Microwaves
Variation of Resistance with Temperature.
Resistive and reactive impedances in RC circuits
Radiation from a Hot Body
Report writing feedback
9
10
19
20
Computer Error Simulations and Analysis
Group Easter Challenge - The strength of paper
11
21
Formal Report writing – no experiments.
8
3.2
Checklist
BEFORE THE LAB SESSION
 Read through the notes on the experiment that you will be doing BEFORE
coming to the practical class. You will be expected to have read all the
introductory notes and refreshed yourself of any knowledge of the subject
taught in school

Think about the safety considerations that there might be associated with the
practical, having read through the lab notes. Write a Risk Assessment before
coming into the lab, to be discussed with your demonstrator at the start of
the session.

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 an Aims statement before coming into the lab, so that you understand
the basics of what you are about to peform.
DURING THE LAB SESSION
 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.Amend your risk
assessment, if required.
9

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/attach any loose paper (eg. graphs, computer print-outs, questionnaires
etc.) into your lab book.
3.3
Experiment 12: 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.
10
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.
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 2 , where a is the size of

diffracting unit and  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.
Experimental set-up and adjustment of the apparatus
The laser - The source of radiation is a 1 mW helium-neon (HeNe) laser which emits a
coherent beam of light of approximately 4 mm 2 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
11
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.
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.
SLIDE 1
SLIDE 2
SLIDE 3
SLIDE 4
SLIDE 5
Diffacting object(s)
One-dimensional diffraction grating.
Double slits
A series of single slits of different widths.
Two-dimensional diffraction grating.
One-dimensional diffraction grating
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  , 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 in m , 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  ? (Hint: use
dimensional analysis to work it out and then refer to the literature to check the correct
equation.)
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
12
easily determine the zeroth order (centre) and as many higher orders as possible. Sketch
and describe the pattern.
Now, by careful experimental measurement it should be possible to determine the
wavelength of the laser light.
The wavelength  of the light from the laser is given by

d sin  m
m
,
(2)
where the angle θm is indicated in Figure 2
Figure 2: Defining d and θm
Because θm is small, sin  m  tan  m 

d x ( m)
D m
x ( m)
, and [2] becomes
D
(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
 , the wavelength of the HeNe laser. Check that your answer is sensible!
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.
13
3.4
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!
14
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;
1
 3kT  2
c

 ,
 m 
where k is the Boltzmann constant. The sound is not propagated exactly at the speed
2
1
2
1
  2
 c2  but at   times it, where  is the ratio of the principal heat capacities of the
3
1
2
gas.
Thus
Csound =  kT 
 m 


1
2
(1)
Measurement of Csound for known T and m therefore enables  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.
Experiment
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?
15
Show that the length of the tube L is related to the wavelength as L = λ/4, 3 λ/4, 5 λ/4, 7
λ/4

i.e. L  2n  1 , where n is an integer .
4
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.
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  from the gradient. Calculate csound from the relation csound = f, 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 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 , 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  you obtain, together with the values given by the kinetic theory
of gases.
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.
16
3.5
Experiment 14: 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 λ= 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.
17
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.
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
-1
magnetic flux density. The wave propagates with a velocity of 3 x 10 8 m s .
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.
18
Electric field
direction of polarised
electromagnetic
radiation
orientation of
polarisation sensitive
detector
θ
Figure 2. Plane polarised radiation incident at an angle θ 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 E0cosθ. Some detectors give an output
that is proportional to the 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, θ,
I = kEo2cos2θ = Iocos2θ.
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.
Experimental
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.
19
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.
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.
Antinode
/2
E
_
Node
Incident wave velocity c
Reflected wave velocity c
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?
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.
20



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 cosθ and cos2θ
dependencies. What do the results imply?
Demonstration of interference effects
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.
A
transmitter
to meter on
receiver
x
A'
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.

21
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).



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 λ/2 (phase difference of π),
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 -π phase shift, however
with this geometry the Gouy effect is reported to result in a further -π/4 phase shift giving
a total of -3π/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.
22



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 λ = 2.8
cm).
(Note: Here due to diffraction, minima are expected at n = d.sin,
where d is the
slit width.)
Appendix
The experimental arrangement is shown in figure 5 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.
Source
L
d
Metal sheet
δ
Detector
Figure 5. 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 2δ where
  (d 2  L2 )1 2  L
Extrema (i.e. maxima and minima) in intensity occur, taking into account the Gouy effect
when:
(m  1) / 2  2(d 2  L2 )1 2  2L  3 / 8
where m is a positive integer. Note half wavelength path lengths give alternating max
and min and so the “extrema”.
23
3.6
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 boiloff 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 coils); Astronomical imaging
(superconducting detectors are used to count 13 billion year old photons) and
transport (MAGLEV trains).
24
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 + (T-273)) ,
(1)
Where R(T) is the resistance at temperature T (in Kelvin), R273 is the resistance at 273K
and 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 2 nd order polynomial fit.
RT = R273 {1 + (T-273) + (T-273) 2},
(2)
where  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.
Experiment
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.
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.
25
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.
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.
Resistance measurements
The magnitudes of the coil and thermistor resistances will be determined using
multimeters set to the ohms range.
Measure RC (the resistance of the copper coil) and RTh (the resistance of the thermistor)
at room temperature.
26
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 RTh).
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.
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 , , 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?
27
3.7
Experiment 16: Resistive and reactive impedances in RC circuits
Outline
 An introduction to an oscilloscope and to electrical circuits
 An introduction to the behaviour of time varying electronic signals in electronic circuits
involving both reactive and resistive impedances, using a series combination of a
resistor and a capacitor.
 The investigation uses an oscilloscope to examine voltage signals for the capacitor
coupled/high pass filter arrangement. This allows the frequency dependence of the
phase angle between current and voltage and the filter performance to be found.
Experimental Skills
 Reinforcement of the use of coaxial leads and circuit construction with breadboards.
 Reinforcement of the use of oscilloscopes for measuring time varying electrical
signals.
 Introduction of oscilloscope techniques for measuring the phase differences between
signals in both Y-t and XY modes.
Wider Applications
 Resistor-capacitors combinations are widely used in electronic circuits as frequency
filters to let through (or pass) either low or high frequency signals, i.e. as low or high
pass filters respectively.
 With inductors in “LCR circuits”, resonance behaviour can occur described by
mathematics that is analogous to mechanical forced, damped oscillatory systems:
This behaviour is extensively covered in 1st year maths and in 2nd year physics labs.
These tuning circuits are what was at the core of the wireless (radio) communication
revolution.
 The visualisation of orthogonal oscillating signals, as seen during this experiment
with the oscilloscope in XY mode, has very close parallels with the different possible
polarisation of light: the analogies of linear, circular and elliptically polarised light are
all produced in this experiment.
28
1 Introduction
Capacitors, like resistors “impede” current flow, although not in the same way:
 A steady voltage applied to a capacitor causes a charge to build up on the plates of a
capacitor eventually preventing further current flow, whilst alternating currents can
flow on and off the plates; hence low frequency signals are impeded but high
frequency signals are not.
 Whereas resistors heat up and so dissipate electrical power (I2R) capacitors do not:
hence their impedance is said to be “reactive” rather than “resistive” (this is the
same for inductors whose impedance is also reactive).
 Whereas current and voltage are in-phase across a resistor they are 90° out of phase
across capacitors (and inductors).
It is the frequency dependence in alternating current (ac) circuits that has lead to
capacitors being widely used in electronic circuits. In analogue filter networks, they help
remove high frequency signals from dc power supplies or remove unwanted direct
current (dc) voltages from ac signals. In resonant circuits they can be used to ‘pick up’
particular frequencies.
1.1 Impedances of resistors and capacitors
The above considerations lead to a distinction: the general term for something that
impedes current flow is called an “impedance” (Z); whereas the impedances of
capacitors (and inductors) are called reactive (X) and of resistors are called resistive (R).
In all cases impedances are measured in ohms and current and voltage are related by
𝑉
𝑍
In addition, of particular relevance here is that the total impedance of a circuit
containing series combination of a resistor (R) and a capacitor (XC) is given by:
𝐼=
𝑍 = 𝑅 + 𝑋𝐶
[1]
[2]
Resistors: A reminder is probably not needed however, the relationship between the
current I through and voltage V across a resistor is I = V/R. If the voltage is varying
sinusoidally (i.e. 𝑉 = 𝑉0 𝑠𝑖𝑛𝜔𝑡, where 𝜔(= 2𝜋𝑓) is the angular frequency) then:
𝐼=
𝑉0 𝑠𝑖𝑛𝜔𝑡
𝑅
[3]
Hence current and voltage are in phase.
Capacitors: The equation that describes the behaviour of capacitors is 𝑄 = 𝐶𝑉 where Q
is the charge on the plates of the capacitor and C is the constant of proportionality to the
voltage across it and is known as its capacitance. In a similar fashion to a resistor the
magnitude of the charge on the capacitor varies in phase with the voltage. However,
here it is the phase difference between current and voltage that is of interest.
Current is given by
𝐼=
29
𝑑𝑄
𝑑𝑉
=𝐶
= 𝐶𝑉0 𝜔𝑐𝑜𝑠𝜔𝑡
𝑑𝑡
𝑑𝑡
[4]
Hence the current leads the voltage by 90° and the magnitude of the reactance is given
by
|𝑋𝐶 | =
|𝑉 |
|𝑉0 𝑠𝑖𝑛𝜔𝑡|
1
=
=
|𝐼 | |𝐶𝑉0 𝜔𝑐𝑜𝑠𝜔𝑡| 𝜔𝐶
[5]
i.e. the reactance of a capacitor decreases with increasing frequency.
1.2 Series RC circuit theory.
Capacitors and resistors often occur in circuits together. In these “RC circuits” the
capacitive reactance and resistance combine to produce an overall circuit. The study of
current and voltage in a series combination of a resistor and a capacitor is the subject of
this experiment.
Consider a sinusoidally varying voltage source connected to a resistor and capacitor in
series as shown in figure 1. The instantaneous voltage across both components must
equal the input voltage and the instantaneous current at all parts of the circuit must be
the same hence equation [6]
𝑉𝑖𝑛 = 𝑉𝑅 + 𝑉𝐶 = 𝐼(𝑅 + 𝑋𝐶 )
[6]
C
VC
Vin
R
VR
Figure 1 Series combination of a resistor and capacitor and the voltages across them.
However, due to the phase differences the voltage across each component peaks at
different times and therefore it is incorrect to add their amplitudes. To understand and
express what is happening it is useful to make use of complex number representations.
An Argand diagram of impedance, Z as shown in Figure 2.
30
ImZ
R
φ
ReZ
-j/ωC
Ztot=R-j/ωC
Figure 2 Argand diagram (similar to phasor diagram) for the impedance of a series RC circuit. The angle 𝜙 is the phase
angle difference between current and input voltage, Vin.
The resistive impedance, R is on the real axis as current and voltage are in phase
(experimentally this is very important – measuring the voltage across any resistor gives
the phase of the current and, if R is known, its magnitude).
By contrast the reactive impedance of the capacitor is given by:
𝑗
[7]
𝜔𝐶
in order to be consistent with the current (which is the same at all parts of the circuit)
leading the voltage across the capacitor by 90°.
𝑗
Using equation 2 the impedance of the series combination of R and C is 𝑍𝑡𝑜𝑡 = 𝑅 − 𝜔𝐶
𝑋𝐶 = −
1
2
The magnitude of the total impedance is given by |𝑍𝑡𝑜𝑡 | = √𝑅 2 + (𝜔𝐶 )
31
1.3 The capacitor coupled, high pass filter arrangement
A common practical use of RC circuits is as “frequency filters”. A voltage signal from one
part of the circuit is passed to the filter (as the filter input signal, Vin) and a different signal
(filter output, Vout = VR) is passed onto the next part of the circuit. With a series
combination of one capacitor and resistor Vin is applied across both components whilst
Vout is taken from either the capacitor or the resistor. Only the latter case will be
investigated in this study and is shown in figure 3. It is known as the “capacitor coupling
arrangement” as the capacitor connects to the circuit that precedes it.
Figure 3 Equivalent arrangements for capacitor coupling/high pass filter.
In this investigation the input signal to the filter Vin will be supplied by a signal generator
and both Vin and Vout will be monitored by an oscilloscope. This arrangement was chosen
since, as discussed previously, the voltage across the resistor (Vout) is the same as, and so
gives, the phase of the current.
From the Argand diagram in figure 2 the phase of the input voltage signal must be between
that across the resistor and capacitor. In addition, the phase angle between input voltage
(across both R and C) and current is given by
1
[8]
𝜔𝑅𝐶
The amplitude of the output voltage can be found by considering the magnitude of the
impedances and considering the circuit as a voltage divider:
𝑡𝑎𝑛𝜙 =
|𝑉𝑜𝑢𝑡 |
=
|𝑉𝑖𝑛 |
𝑅
2
√𝑅2 + ( 1 )
𝜔𝐶
=
𝜔𝑅𝐶
((𝜔𝑅𝐶 )2 + 1)1⁄2
[9]
Filter characteristics as a function of frequency, remembering that RC is the time
constant of the circuit, are summarised in table 1.
Table 1 Filter characteristics as a function of frequency
Output signal, |𝑽𝒐𝒖𝒕 |
Phase angle, 𝝓
|𝑉𝑜𝑢𝑡 | → 0
𝜙 → 900
𝜔 = 1⁄𝑅𝐶
|𝑉𝑜𝑢𝑡 | = |𝑉𝑖𝑛 |⁄√2
𝜙 = 450
𝜔 ≫ 1⁄𝑅𝐶 (high frequency)
|𝑉𝑜𝑢𝑡 | → |𝑉𝑖𝑛 |
𝜙 → 00
Frequency, 𝝎
𝜔 ≪ 1⁄𝑅𝐶 (low frequency)
32
At low frequencies the impedance of the capacitor dominates and most of the input
voltage is dropped across it, whereas at high frequencies the reverse is true. This is why
the arrangement is known as a high pass filter: the input signal is only passed on faithfully
(i.e. without attenuation) at high frequencies.
Aside: A “tweeter” is the loudspeaker in audio systems that is designed to generate high
frequency sound (f > 2 kHz typically). High pass filters very similar to the one measured
here are used to ensure that only the high frequencies are delivered to the tweeter.
2 Experimental
Using the prototype board, assemble the circuit in Figure 3 making use of three coaxial
leads and connector posts and ensuring that:
 When connecting jump leads to the 4 mm posts ensure that some bare wire protrudes
from the post: a common cause of poor connections is pinching down on the wires’
insulation.
 The earth of the three coaxial leads join at the same post (otherwise they will short
out voltage signals).
 The input and output signals are taken to Ch1 and Ch2 of the oscilloscope respectively.
 The function generator is set to sine wave and its “dc offset” is turned off.
The capacitor and resistor provided have nominal values of 0.022 F and 4.3 kΩ
respectively. However, measure the resistor value with a multi-meter and use this later
to find the value of the capacitor (the quoted tolerance on the value given is 10%).
With the circuit made up, get used to operating the oscilloscope. Reminder: a summary
version of how to use the oscilloscope can be found in background notes. But to start:
 Turn on the oscilloscope and when the GW Instek banner has disappeared press
“Save/Recall” then select “Default Setup” and finally press “Autoset”.
 Or you could simply press “Autoset” – but this may (if you are very unlucky) remember
unsuitable previous conditions.
Now:
 Adjust the signal generator to set an input signal (dc offset in off position) with a peak
to peak amplitude of ~3 V.
 Use the vertical adjustments on Ch1 and 2 so that they are both at 0V (the position
appears at the bottom left of the trace as they are being adjusted) to make phase and
signal changes more obvious.
 Check that the circuit is working as expected, i.e. that as the input signal frequency is
varied the output signal size and phase vary roughly as described at the end of section
1.3.
Note: the same circuit arrangement will be used for all subsequent measurements. If you
are unsure that it is working correctly check with a demonstrator.
2.1 Measuring the filter characteristics
With the set up as above, and with the time base and y scales adjusted as appropriate
perform measurements of frequency, f (and so period, T=1/f), Vin (although this isn’t
33
adjusted it may drift so measure it), Vout and the lead or lag of one oscillation against the
other, dT (and so the phase offset 𝜙).
𝑑𝑇
𝑑𝑇
degrees (or 2𝜋
radians)
𝑇
𝑇
As 1 period, T, corresponds to 1 cycle, 360°, 2π radians.
Note:

𝜙 = 360
Do this as a function of frequency (take ~10 readings in the range 200 Hz to 8000 Hz)
recording the results in a suitable table.
Most measurements are made by the oscilloscope and can be read from its display using
its “measure” facility (use peak to peak amplitudes for voltages), but for dT use the two
X cursors (and then convert to radians or degrees as required). It will be necessary to
toggle between “cursor” and “measure”.

Make plots of phase angle and |𝑉𝑜𝑢𝑡 |/|𝑉𝑖𝑛 | versus frequency, use these to find the
condition 𝜔 = 1⁄𝑅𝐶 and so determine the value of the capacitance (see table 1).
Using the phase angle data plot a suitable straight line graph (see equation 8), use this
to determine the capacitance and compare the value with that above.

2.2 Using the oscilloscope XY mode for determination of filter characteristics.
Here the x axis is not time dependent, instead one of the two channel inputs produces x
deflections and the other y. This mode will be used to repeat the measurements of the
previous section, but first some explanation.
The movement of the spot on the screen is then described by
𝑥 = 𝐴𝑠𝑖𝑛(𝜔𝑡);
𝑦 = 𝐵𝑠𝑖𝑛(𝜔𝑡 − 𝜙)
[10]
where  is the phase angle between the 2 inputs. In general this represents an ellipse, as
shown in Figure 4, although depending on the phase angle the ellipse may appear when
in-phase as a straight line through to, with A = B and 90° out of phase, a perfect circle.
Such plots are known as Lissajous plots or figures.
Figure 4: Elliptical trace for the measurement of phase angle. Also shown dotted are straight line (𝜙=0) and circular
(𝜙 = 90°, 𝐴 = 𝐵) traces.
Understanding the XY mode
To understand what you are seeing do the following (you will almost certainly need to
get help from a demonstrator to get you started here):
34



Sketch one period of a time varying sine wave and a cosine wave, both of amplitude
1, in your diary. On both mark 10 reasonably evenly time spaced points and
number these from 0 to 9 (point at start and end of cycle numbered 0 and 9
respectively).
Draw an XY plot with scales -1 to +1 in both X and Y. On this and for the case when
both X and Y vary sinusoidally, plot out the time progression of the display using the
numbers 0 to 9 as markers (rather than x’s or o’s). This is a Lissajous figure for the
case of signals of the same frequency and in phase.
Repeat for X a sine wave and Y a cosine wave. This is the case of X and Y 90° out of
phase. Using the time progression note whether the resulting (hopefully circular)
trace was drawn out in a clockwise or anticlockwise sense.
Analysis of plots such as figure 4 (to find both |𝑽𝒐𝒖𝒕 |/|𝑽𝒊𝒏 | and 𝝓).
To find 𝜙 the line y = 0 (passing through A’,N’,O,N and A) through the ellipse is
considered.
We
have
Hence
𝑦 = 𝐵𝑠𝑖𝑛(𝜔𝑡 − 𝜙);
So that 𝜔𝑡 = 𝜙
𝑥 = 𝐴𝑠𝑖𝑛(𝜔𝑡) = 𝐴𝑠𝑖𝑛𝜙 = 𝑁 = |𝑁′|
[11]
[12]
𝑁 𝑁𝑁′
[13]
=
𝐴 𝐴𝐴′
Here AA' is the difference length between the two extreme x values of the ellipse, and NN'
is the length given by the intersection of the ellipse with the x axis. Using the cursors it is
more convenient to obtain these from the oscilloscope trace than N and A.
And
𝑠𝑖𝑛𝜙 =
If the input signal (Vin) to a circuit (here the signal from the signal generator) is applied to
channel 1 (X) and the output signal (Vout) from the circuit (here from across the resistor)
to channel 2 then from figure 4 we have:
|𝑉𝑜𝑢𝑡 | 𝐵 𝑌𝑝𝑝 𝐶ℎ2𝑝𝑝
= =
=
|𝑉𝑖𝑛 |
𝐴 𝑋𝑝𝑝 𝐶ℎ1𝑝𝑝
Where the pp subscript indicates peak to peak amplitude as the oscilloscope finds in its
“measure” mode.
Measurements
To put the scope in XY mode press the “menu” button under horizontal and then select
XY.
 Make measurements of |𝑉𝑜𝑢𝑡 |/|𝑉𝑖𝑛 |and phase offset 𝜙 versus frequency.
 Add this data to your earlier plots ($2.1) of phase angle and |𝑉𝑜𝑢𝑡 |/|𝑉𝑖𝑛 | versus
frequency and comment on the agreement.
3 Conclusions
As part of your concluding remarks consider the relative merits of the different methods
for measuring phase offsets and determining C.
35
Appendix: Complex number treatment of output voltage across the resistor
We are dealing with a potential divider circuit in which (using complex ohm’s law)
𝑉𝑖𝑛 = 𝐼 (𝑅 −
𝑗
)
𝜔𝐶
And, since the current must be the same in all parts of the circuit
𝐼=
𝑉𝑜𝑢𝑡
𝑉𝑖𝑛
=
𝑗
𝑅
(𝑅 − 𝜔𝐶 )
Rearranged this gives
𝑉𝑜𝑢𝑡 = 𝐼𝑅 =
𝑉𝑖𝑛 𝑅
𝑗
(𝑅 − 𝜔𝐶 )
=
𝑗
𝑗
(𝑅 + 𝜔𝐶 )
𝑉𝑖𝑛 𝑅
𝑉𝑖𝑛 𝑅 (𝑅 + 𝜔𝐶 )
.
=
2
𝑗
𝑗
(𝑅 − 𝜔𝐶 ) (𝑅 + 𝜔𝐶 ) (𝑅2 + ( 1 ) )
𝜔𝐶
The amplitude of the output (usually of most interest) is given by:
1⁄2
(𝑉𝑖𝑛 𝑅
∗ )1⁄ 2
|𝑉𝑜𝑢𝑡 | = (𝑉𝑜𝑢𝑡 𝑉𝑜𝑢𝑡
=
=
(
𝑉𝑖𝑛 𝑅
(𝑅2 + (
1
)2
2
1
1
2 2
(𝑅 + 𝜔2 𝐶 2)
(𝑅2 + (
) )
𝜔𝐶
2 1⁄ 2
) )
𝜔𝐶
)
𝑉𝑖𝑛 𝜔𝑅𝐶
=
((𝜔𝑅𝐶 )2 + 1)1⁄2
Phasor diagrams
These are an alternative to the Argand diagram of figure 1 to represent such time
varying voltages and currents and are used in Halliday and Resnick (chapters 16 and 31).
Figures may appear very similar to figure 2, however, the vectors rotates anticlockwise
with constant angular velocity corresponding to the angular frequency of the quantity
involved. The length of the vectors is equal to the amplitude of the quantity and the
instantaneous value of a quantity is represented by the projection onto the vertical axis.
36
3.8
Experiment 17: Radiation from a Hot Body
Outline
A body (a light bulb filament) is heated by passing an electrical current through it. Knowing
the voltage and current and something about the filament it is possible to determine the
energy lost as a function of its temperature. This very simple experiment provides a good
introduction to electrical circuits and measuring techniques, the manipulation of
equations into a suitable form for analysis and the introduction of logarithmic plots in
determining so-called power law dependencies. The physics in this experiment is not
covered in lecture courses in autumn semester but will appear in the spring (in PX0201
and PX0202).
Introduction
A body at an absolute temperature T (T(K) = T(oC) + 273) placed in a vacuum will lose heat
by radiation. Assuming there are no radiation sources surrounding the body, the body
would eventually cool down to absolute zero (T=0 Kelvin). The rate of loss of heat by
radiation for a body placed in a vacuum is proportional to T4. For a perfect emitter (or
black body)
𝑊 = 𝜎𝐴𝑇 4
(1)
where A is the area of the radiating surface of the body (in m ),  is Stefan's constant (5.6
x 10-8 W m-2 K-4 ) and W is the rate of heat loss of the body (in watt). The above equation
is called the Stefan-Boltzmann law. In practice, most bodies cannot be considered as
perfect emitters and the Stefan-Boltzmann equation has to be written
𝑊 = 𝜀𝜎𝐴𝑇 4
(2)
where the emissivity factor  is a number less than unity (and which depends upon the
material). However, the surroundings have an ambient temperature T0 and so the net
rate of heat loss of the body is given by the difference at which it loses heat by radiation
and absorbs heat from its surroundings by radiation, so that,
𝑊 = 𝜀𝜎𝐴(𝑇 4 − 𝑇04 )
(3)
Eventually, if no energy is supplied to the body to make up for the radiative heat loss, the
body will cool down until at T  T0 dynamic equilibrium exists in which energy is
continually being exchanged with the surroundings, but the net heat loss is zero.
Experiment
Note: There is a lot of number crunching at the end of this experiment so it is important
to carry out the measurements efficiently.
In this experiment, the energy radiated from the tungsten filament of a small electric light
bulb is measured as a function of temperature. This is done by passing a current through
the filament, waiting until the temperature of the filament has stabilised, and then
measuring both the current I through the filament and the voltage V across it. (Note that
the filament is not in thermal equilibrium with its surroundings since its temperature is
always higher than that of the surroundings (ie T  T0 , except when no current flows.
However it is in a steady state, in which the power supplied to the filament by the power
37
supply is exactly equal to the radiative heat loss, so that the temperature is constant and
measurements can easily be made).
The power dissipated in the bulb can be found from the voltage V across it and the current
I through it:
𝑊 = 𝐼𝑉
(4)
The temperature of the filament can be found using the formula,
𝑅 = 𝑅0 [1 + 𝛼 (𝑇 − 𝑇0 )]
(5)
where R0 is the value of the resistance R of the filament at room temperature T0 and  is
the average temperature coefficient of resistance (= 0.0053 K-1 for tungsten). By
measuring R (using R = V/I) it is easy to calculate T.
The above information is then used to check the relation, equation (1).
Method
You are provided with a small electric light bulb (25 V) mounted in a box, a DC power
supply, a voltmeter and an ammeter. Make the connections as indicated on the box. The
circuit is then equivalent to the one shown below:
Power
supply
ammeter
voltmeter
bulb
Figure 1. Circuit diagram
DO NOT EXCEED 20 VOLTs

38
STEP 1: R0 can be obtained from the limiting value of R as I  0 . Unfortunately, whilst
it is important to get an accurate value for R0 this approach will not usually let you do
this (mainly due to the inaccuracy of the current meter at low currents). The
alternative is to directly measure R 0 (the resistance of the filament when it is cold)
using the voltmeter connections on the box and the multi-meter as an ohm-meter
(disconnect all other wires when doing this). Use this value in your calculations and
compare it with the resistances you later calculate for small currents. Clearly R 0 should
be the smallest resistance value since the bulb is at its coldest for this measurement.
There is a thermometer at the front of the laboratory that will give you room
temperature.

STEP 2: Take readings of V and I (the expected maximum current is ~75 mA = 0.075 A).
You should tabulate your data, including columns for I and V data and calculated values
of power W (=I.V), resistance R (=V/I), temperature T (in Kelvin), T4 and T4 - T04.
Making sure that V DOES NOT EXCEED 20 volts, take readings in ~2 V steps up to 20
V.
Note - when you change the temperature of the filament let the system reach
equilibrium before taking a reading. In addition, note the conditions under which the
filament starts to glow so that you can estimate the temperature of the filament at
this point.

STEP 3: Tabulate values of log10W and log10T and plot a graph of log10W versus log10T.
If W  T n then a log-log plot should produce a straight line of gradient n* (see “maths”
below). From your graph find n (and its error) to see if a T4 dependence is observed.
Try to explain any discrepancy.
Note: In the following analysis you will not be required to work out errors as there is
unlikely to be sufficient time.

STEP 4: Plot W versus T4 - T04. Using equation 3, calculate the total surface area A of
the filament (assume that  = 0.1). Note there will be a systematic error introduced
here if the temperature dependence isn’t T4.
*Maths
If we have
Taking log10 of both sides
𝑊 = 𝑘𝑇 𝑛
log10 𝑊 = log10 (𝑘𝑇 𝑛 ) = log10 𝑇 𝑛 = log10 𝑘 = 𝑛log10 𝑇 + log10 𝑘
so a graph of log10W against log10T will be a straight line of slope n.
39
3.9
Experiment 18: Report writing feedback
Aim: This session is primarily to provide you with feedback for your formal report that you
submitted before Christmas. You will sit with the session supervisor and go through
feedback.
However it is also a chance to see what is involved in assessment of reports by actually
having a chance to peer review a number of examples as an exercise, and then in your
small peer group will be able to discuss the issues associated with good science
communication and why certain conventions adopted by scientists are important.
40
3.10 Experiment 19: 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.
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”.
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.
41
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.
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.
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.
42
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).
Addition of A and B (Sum, S=A+B)
Reminder: error propagation for P = A + B
Partial differentiation gives
𝜕𝑆
𝜕𝐴
=1
𝜕𝑆
and
𝜕𝐵
=1
Or
𝜕𝑆 = 𝜕𝐴
and 𝜕𝑆 = 𝜕𝐵
Combining the 𝜕𝑃 (or ΔP) contributions in quadrature gives the familiar
(∆𝑆)2 = (∆𝐴)2 + (∆𝐵)2
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
𝜕𝑃
𝜕𝐴
=𝐵
𝜕𝑃
and
Or
𝜕𝑃 = 𝐵𝜕𝐴
and
Combining the 𝜕𝑃 (or ΔP) contributions in quadrature
𝜕𝐵
=𝐴
𝜕𝑃 = 𝐴𝜕𝐵
(∆𝑃)2 = (𝐵∆𝐴)2 + (𝐴∆𝐵)2
Dividing by P2 = (AB)2 gives the familiar
∆𝑃 2
∆𝐴 2
∆𝐵 2
(𝑃) = (𝐴) +(𝐵)
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.
43

Compare the mean of S with the expected value and its standard deviation with the
error in S calculated in the usual way.
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.
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.
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
0
0.1
0.2
44
Length (y_data)/m
0.055
0.074
0.089
0.4
0.5
0.6
0.8
0.124
0.135
0.181
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 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.
45




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.
46
3.11 Experiment 20: Group Easter Challenge - The strength of paper
?
You will be assigned into a group for one lab session where you will undertake a team
task. This will, like Rube Goldberg, be a competitive test against your fellow students.
47
4
Error propagation: A reminder of the general case
The problem here is to find the overall change of a function due to (small) changes in its
component parts. The answer can be found using calculus, if a value z is a function of x
and y, (i.e. z = f(x,y)) partial differentiation can be used to find the effect of a small change
in either x or y. (Partial differentiation is taught in the first year and the process is
essentially one of differentiating with respect to (w.r.t.) one variable whilst holding all the
others constant).
The partial differential of z with respect to x (holding y constant) is given by z x so that
the change in z (i.e. Δz )due to a small change in x (i.e. Δx) is:
z
z 
x
x
[3]
There is a similar expression for changes in z due to changes in y and the total change in z,
i.e. the “total differential” is then given by
z 
z
z
x  y
x
y
[4]
The above equation concerns two variables but clearly the number of terms on the right
hand side would increase to match the number of variables in an arbitrary function. Even
so, Δz in the above equation cannot be used as the combined error arising from the errors,
Δx and Δy, in x and y respectively. The reason is that in the above equation the signs of
both the derivatives and the errors are important. As presented then the signs of multiple
terms (2 here) could lead to the situation where two large but opposite contributions
cancel each other, resulting in an underestimated error.
One way to resolve this issue would be to add the magnitudes of the terms on the rhs of
the equation. However, this is equivalent to having the errors contribution due to x and y
always reinforcing each other which is not realistic either. Instead, the conventional
solution is to square all of the terms, i.e.:
2
2
 z 
 z 
(z )    x 2    y 2
 x 
 y 
2
[5]
Δz in this equation is the overall error. The resulting errors are realistic and are often said
to have been combined in “quadrature” (quadrature is often used to mean squaring).
Example. Resistance, R = f(V,I) = V/I.
The aim is to show how the overall error for resistance is found using the values and errors
for voltage and current.
First consider the total derivative
R
R
1
V
R
R
R 
V 
I  V 
I  V  I
2
V
I
I
V
I
I
Rearranging
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R V I


R
V
I
2
2
2
 R 
 V 
 I 
Squaring each term 
 
  
 R 
 V 
 I 
This methodology used here for a quotient can be used generally and the more common
results are given in the next section
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5
Lab Diary 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)
6
Formal 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.
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