and Electromagnetism Optics
The Lab Manual for PHY 103N
Engineering Physics II Laboratory
Department of Physics
University of Texas at Austin, Austin, TX 78712
2007-2008
November 14, 2008
ii
Contents
Preface
ix
0 Introduction
0.1 General Lab Procedures . . . . . . . . . . . . . . . . .
0.2 Error Estimation and Propagation . . . . . . . . . . . .
0.2.1 Denition of Uncertainty . . . . . . . . . . . . .
0.2.2 Estimating Parameters and Their Uncertainties
0.2.3 Propagating and Reporting Uncertainties . . . .
0.3 The Lab Worksheet . . . . . . . . . . . . . . . . . . . .
0.3.1 Purpose . . . . . . . . . . . . . . . . . . . . . .
0.3.2 Procedure . . . . . . . . . . . . . . . . . . . . .
0.3.3 Computer Work . . . . . . . . . . . . . . . . . .
0.3.4 Pre-Classroom Checklist . . . . . . . . . . . . .
0.3.5 Calculations & Analysis . . . . . . . . . . . . .
0.3.6 Discussion and Conclusion . . . . . . . . . . . .
0.W1Error Analysis Worksheet . . . . . . . . . . . . . . . .
0.W1.1 Purpose . . . . . . . . . . . . . . . . . . . . . .
0.W1.2 Procedure . . . . . . . . . . . . . . . . . . . . .
0.W1.3 Computer Work . . . . . . . . . . . . . . . . .
0.W1.4 Enter Data . . . . . . . . . . . . . . . . . . . .
0.W1.5 Entering Formulas . . . . . . . . . . . . . . . .
0.W1.6 Plotting Data . . . . . . . . . . . . . . . . . . .
0.W1.7 Performing a Weighted Least-Squares Fit . . . .
0.W1 In-Classroom Calculations . . . . . . . . . . . . . . .
0.W2KaleidaGraph & Graphing By Hand . . . . . . . . . .
0.W2.1 Purpose . . . . . . . . . . . . . . . . . . . . . .
0.W2.2 Procedure . . . . . . . . . . . . . . . . . . . . .
0.W2.3 Computer Work . . . . . . . . . . . . . . . . . .
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iv
CONTENTS
0.W2.4 Calculation & Analysis . . . . . . . . . . . . . . . . . . 37
1 Electrostatics
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Electric Charge . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Coulombs Law . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Electric Field . . . . . . . . . . . . . . . . . . . . . . .
1.2.4 Gauss Law . . . . . . . . . . . . . . . . . . . . . . . .
1.2.5 Electric Potential . . . . . . . . . . . . . . . . . . . . .
1.3 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 The Pasco ES-9077 Electrostatics Voltage Source . . .
1.3.2 The Pasco ES-9078 Electrometer . . . . . . . . . . . .
1.3.3 The Pasco ES-9042A Faraday Ice Pail . . . . . . . . .
1.3.4 The Pasco ES-9075A Charge Producers and Proof Planes
1.3.5 The Pasco ES-9059 13-cm Spheres . . . . . . . . . . .
1.3.6 The Pasco High Resistance Paper and Conductive Ink
1.W Electrostatics Worksheet . . . . . . . . . . . . . . . . . . . . .
1.W.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . .
1.W.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.3 Computer Work . . . . . . . . . . . . . . . . . . . . . .
1.4.4 Pre-Classroom Check List . . . . . . . . . . . . . . . .
1.4.5 Classroom Calculations & Analysis . . . . . . . . . . .
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2 DC Circuits
2.1 Introduction . . . . . . . . . . . . . . . . . . . . .
2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 References . . . . . . . . . . . . . . . . . .
2.2.2 Voltage, Current, and Resistance . . . . .
2.2.3 Diagrams and Meters . . . . . . . . . . . .
2.2.4 Voltage Sources . . . . . . . . . . . . . . .
2.2.5 Series and Parallel Circuits with Resistors
2.2.6 The Color Code for Resistors . . . . . . .
2.3 Apparatus . . . . . . . . . . . . . . . . . . . . . .
2.3.1 The Multimeter . . . . . . . . . . . . . . .
2.3.2 The Power Supply . . . . . . . . . . . . .
2.3.3 The Breadboard . . . . . . . . . . . . . . .
2.W1DC Circuits Part I Worksheet . . . . . . . . . . .
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v
CONTENTS
2.W1.1 Purpose . . . . . . . . . . . . . .
2.W1.2 Procedure . . . . . . . . . . . . .
2.W1.3 Part I Pre-Classroom Check List
2.W1.4 In-Classroom Questions . . . . .
2.W2DC Circuits Part II Worksheet . . . . . .
2.W2.1 Purpose . . . . . . . . . . . . . .
2.W2.2 Procedure . . . . . . . . . . . . .
2.W2.3 Computer Work 1 . . . . . . . . .
2.W2.4 Computer Work 2 . . . . . . . . .
2.W2.5 Pre-Classroom Check List . . . .
2.W2.6 Classroom Questions . . . . . . .
3 Electron Dynamics
3.1 Introduction . . . . . . . . . . . .
3.2 Theory . . . . . . . . . . . . . . .
3.2.1 References . . . . . . . . .
3.2.2 The Physical Situation . .
3.2.3 Uniform Electric Field . .
3.2.4 Uniform Magnetic Field .
3.3 Apparatus . . . . . . . . . . . . .
3.W Electron Dynamics Worksheet . .
3.W.1 Purpose . . . . . . . . . .
3.W.2 Procedure . . . . . . . . .
3.W.3 Computer Work . . . . . .
3.4.4 Pre-classroom Check List .
3.4.5 Calculations & Analysis .
3.4.6 Discussion and Conclusion
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4 Measurements with the Oscilloscope
4.1 Introduction . . . . . . . . . . . . . .
4.2 Theory . . . . . . . . . . . . . . . . .
4.2.1 References . . . . . . . . . . .
4.2.2 The Properties of Waves . . .
4.3 Apparatus . . . . . . . . . . . . . . .
4.3.1 The Oscilloscope . . . . . . .
4.3.2 The Function Generator . . .
4.3.3 The Phase Shifter . . . . . . .
4.W Oscilloscope Worksheet . . . . . . . .
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103
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vi
CONTENTS
4.W.1
4.W.2
4.W.3
4.W.4
Purpose . . . . . . . . . .
Procedure . . . . . . . . .
Pre-Classroom Check List
Calculations & Analysis .
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5 RC Circuits and Filters
5.1 Introduction . . . . . . . . . . . . . . . .
5.2 Theory . . . . . . . . . . . . . . . . . . .
5.2.1 References . . . . . . . . . . . . .
5.2.2 DC Response of Capacitors . . .
5.2.3 AC Response of Capacitors . . .
5.3 Apparatus . . . . . . . . . . . . . . . . .
5.W RC Circuits and Filters Worksheet . . .
5.W.1 Purpose . . . . . . . . . . . . . .
5.W.2 Procedure - DC Response . . . .
5.W.3 Computer Work - DC Response .
5.W.4 Procedure - Low-Pass Filter . . .
5.W.5 Computer Work - Low-Pass Filter
5.W.6 Procedure - High-Pass Filter . .
5.W.7 Pre-Classroom Check List . . . .
5.W.8 Calculations and Analysis . . . .
5.W.9 Discussion and Conclusion . . . .
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6 Electromagnetic Induction
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
6.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 References . . . . . . . . . . . . . . . . . . .
6.2.2 Magnetic Flux . . . . . . . . . . . . . . . . .
6.2.3 Faradays Law . . . . . . . . . . . . . . . . .
6.2.4 The Transformer . . . . . . . . . . . . . . .
6.3 Apparatus . . . . . . . . . . . . . . . . . . . . . . .
6.W Electromagnetic Induction Worksheet . . . . . . . .
6.W.1 Purpose . . . . . . . . . . . . . . . . . . . .
6.W.2 Lab Work . . . . . . . . . . . . . . . . . . .
6.W.3 Computer Work . . . . . . . . . . . . . . . .
6.W.4 Triangle and Square Wave Input Lab Work
6.W.5 Pre-Classroom Check List . . . . . . . . . .
6.W.6 Calculations and Analysis . . . . . . . . . .
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135
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145
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169
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vii
CONTENTS
6.W.7 Discussion and Conclusion . . . . . . . . . . . . . . . . 184
7 Polarization of Light
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
7.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1 References . . . . . . . . . . . . . . . . . . . .
7.2.2 Polarization . . . . . . . . . . . . . . . . . . .
7.2.3 Polarizers . . . . . . . . . . . . . . . . . . . .
7.2.4 Malus Law . . . . . . . . . . . . . . . . . . .
7.3 Apparatus . . . . . . . . . . . . . . . . . . . . . . . .
7.W Polarization of Light Worksheet . . . . . . . . . . . .
7.W.1 Purpose . . . . . . . . . . . . . . . . . . . . .
7.W.2 Procedure - Polarizers and Intensity . . . . . .
7.W.3 Computer Work - Polarizers and Intensity . .
7.W.4 Procedure - Polarized Light and a Polarizer .
7.W.5 Computer Work - Polarized Light & Polarizer
7.W.6 Procedure - Crossed Polarizers . . . . . . . . .
7.W.7 Computer Work - Crossed Polarizers . . . . .
7.W.8 Pre-Classroom Check List . . . . . . . . . . .
7.W.9 Calculations and Analysis . . . . . . . . . . .
7.W.10 Discussion and Conclusion . . . . . . . . . . .
8 Refraction Optics
8.1 Introduction . . . . . . . . . . . . . . . .
8.2 Theory . . . . . . . . . . . . . . . . . . .
8.2.1 References . . . . . . . . . . . . .
8.2.2 Index of Refraction . . . . . . . .
8.2.3 Refraction at a Boundary - Snells
8.2.4 Total Internal Reection . . . . .
8.3 Apparatus . . . . . . . . . . . . . . . . .
8.W Refraction Worksheet . . . . . . . . . . .
8.W.1 Purpose . . . . . . . . . . . . . .
8.W.2 Procedure . . . . . . . . . . . . .
8.W.3 Pre-Classroom Check List . . . .
8.W.4 Calculations and Analysis . . . .
8.W.5 Discussion . . . . . . . . . . . . .
8.W.6 Conclusion . . . . . . . . . . . . .
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185
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201
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viii
CONTENTS
9 Imaging Optics
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.1 References . . . . . . . . . . . . . . . . . . . . .
9.2.2 Imaging Optics and Lenses . . . . . . . . . . . .
9.2.3 Compound Lenses . . . . . . . . . . . . . . . .
9.2.4 The Thin Lens Equation . . . . . . . . . . . . .
9.2.5 Magnication . . . . . . . . . . . . . . . . . . .
9.3 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . .
9.W Imaging Optics Worksheet . . . . . . . . . . . . . . . .
9.W.1 Purpose . . . . . . . . . . . . . . . . . . . . . .
9.W.2 Procedure - Focal Length and Magnication . .
9.W.3 Computer Work -Focal Length & Magnication
9.W.4 Procedure - Compound Lens System . . . . . .
9.W.5 Pre-Classroom Checklist . . . . . . . . . . . . .
9.W.6 Calculations and Analysis . . . . . . . . . . . .
9.W.7 Discussion . . . . . . . . . . . . . . . . . . . . .
9.W.8 Conclusion . . . . . . . . . . . . . . . . . . . . .
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10 Diraction and Interference of Light
10.1 Introduction . . . . . . . . . . . . . . . . .
10.2 Theory . . . . . . . . . . . . . . . . . . . .
10.2.1 References . . . . . . . . . . . . . .
10.2.2 Diraction and Interference . . . .
10.2.3 Two-Slit Interference . . . . . . . .
10.2.4 Multiple-Slit Interference . . . . . .
10.2.5 Diraction Grating . . . . . . . . .
10.2.6 Single-Slit Diraction and Available
10.2.7 Multiple-Slit Diraction . . . . . .
10.3 Apparatus . . . . . . . . . . . . . . . . . .
10.WDiraction and Interference Worksheet . .
10.W.1 Purpose . . . . . . . . . . . . . . .
10.W.2 Procedure . . . . . . . . . . . . . .
10.W.3 Computer Work . . . . . . . . . . .
10.W.4 Pre-Classroom Check List . . . . .
10.W.5 Calculations & Analysis . . . . . .
10.W.6 In-Classroom Discussion . . . . . .
10.W.7 In-Classroom Conclusion . . . . . .
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Preface
Welcome to Physics 103N, Engineering Physics II Lab. This class is the continuation of Physics 103M, and is a corequisite to Physics 303L, Engineering
Physics II Lecture. Although this class is a corequisite to Physics 303L, the
topics we discuss here are not exactly those discussed in lecture. There are
several reasons for this: the rst is that timing the labs with the lectures is
very dicult. Second, you dont always need a detailed theoretical description of phenomena to measure and characterize their properties. It is this
empirical approach that we want to emphasize here. Third, because of time,
there are important physical phenomena that are not covered in detail in the
lecture, and we will examine some of these more closely.
There are two essential reasons for this course. First, it will give general
background knowledge of how to do experimental work properly. You will
learn how to use equipment such as multimeters, frequency generators, and
oscilloscopes among others. Further, you will see how to measure various
properties of electronic circuits and optical systems and learn the importance
of experimental error when doing scientic work. These are all very practical
skills, and remember, engineering is experimental work.
Secondly, it will help you see that all the conjectures and calculations
learned in lecture do describe events in the real world. You will learn how to
empirically test a hypothesis - the very basis of science. We will be able to
verify some of the formulas derived in the lecture as a check on what the professor has been teaching. If these theories can be veried with experiments,
then you will probably believe what else is said in lecture. Whereas if they
cannot be veried, that makes everything else the professor expounds liable
to suspicion. So be on the lookout for discrepancies!
Most of the equipment you need will be provided in lab. You should
bring a pen and pencil and paper, a scientic calculator (i.e., one with logs
ix
and trigonometric functions, not necessarily a graphing calculator), and this
manual to each lab meeting. You might want to keep a notebook to record
your data and notes for the class. Your reports will be turned in on the
worksheets printed in this manual, but extra paper may be necessary from
time to time. So, if you do use a notebook, make sure it has perforated
sheets. In any case, avoid the hardbound laboratory notebooks, since they are
unnecessarily expensive (>$10). We also expect that you have the textbook
assigned to the 303L lecture course available; the reference is
R. A. Serway and J. W. Jewitt, Physics for Scientists & Engineers, 6th edition (updated), Thomson Publishing, Belmont, CA
(2004).
An additional reference, that well refer to repeatedly in our discussion of
error analysis in Chapter 0, but is by no means required reading, is
P. R. Bevington and D.K. Robinson, Data Reduction and Error
Analysis for the Physical Sciences, 2nd. edition, McGraw-Hill,
Inc., New York (1992).
Finally, this lab manual has just been substantially edited. We believe
that these edits have signicantly improved the manual, but some typos, ambiguities, or other inadequacies are bound to have slipped our grasp. Please
bring any errors or confusing parts of the manual to the attention of your
instructor. Student input is invaluable to the production of a document that
students depend on for learning. As an alternative, feel free to E-mail your
comments and suggestions to [e-mail].
Chapter 0
Introduction
This introductory chapter describes general information regarding laboratory
procedures, how to write a lab worksheet and how to analyze data correctly.
Read this information carefully. The instructor will only cover these items
quickly on the rst day of lab; the expectation is that students are already
familiar with the basics by having read this rst chapter ahead of time.
0.1
General Lab Procedures
Physics 103N is a self-contained three hour lab. All work will be completed
during the lab period, so the only homework is to prepare for the next lab
including the weekly prelab. However, this preparation is absolutely essential
for a student to do well. The basic lab format is as follows:
The instructor will spend about 15-20 minutes introducing the lab and
providing a demonstration of how it works.
Students spend two hours collecting data and plotting graphs in the
laboratory. This is a group eort of lab partners.
After the second hour, everyone moves to a classroom to INDIVIDUALLY complete the lab worksheet, which includes calculations, short
answer questions, and frequently, a conclusion.
The worksheet is turned in at the end of the third hour regardless of whether it is nished.
1
2
CHAPTER 0. INTRODUCTION
So the moral of this story is, What is good is also what is bad. On the
one hand, there are no long lab write-ups after taking data, but on the other
hand, each class involves pressure to perform the experiment, analyze data,
and write-up ndings in a three-hour period. Some students are well suited
for this format, but for others, having no time to reect and process the data
really aects their performance. Preparation is the solution. Carefully
read the manual before coming to class, and understand the theories, purpose and procedures for the lab.
Finally, since the worksheet is turned in at the end of class, and this requires preparation to succeed, the instructor may deem quizzes unnecessary,
but the possibility of a quiz exists - another reason to prepare.
0.2
Error Estimation and Propagation
Error is everywhere, and we must not only acknowledge it, but understand it
and control it. Webster denes error as the dierence between an observed
or calculated value and the true value. The problem with this denition is
that we usually have no idea what the true answer should be. Previous
experiments or theoretical calculations may give us a clue, but somehow we
must extract an estimate of the true value from our data. In addition, we
should also determine to what extent we should take our answer seriously.
This last point embodies what we call error estimation.
An example will show why this process is so important. Suppose you
measure the acceleration of a free-falling body and the answer you obtain is
10.5 m/s2 . Have you contradicted the accepted value of 9.7990 0.0014 m/s2 ?
The answer depends on the size of the error in your answer. If your result
was 10.5 1.0 m/s2 then your answer is no, because the accepted value
lies within your error. If, on the other hand, you had performed a fairly
precise measurement and obtained 10.5 0.1 m/s2 , then your answer would
have to be yes; you should then start looking for what went wrong in the
experiment itself, since many, many other measurements conducted over the
last 300 years have established the accepted value 9.7990 0.0014 m/s2 . So,
the amount of eort and trust you put into an answer depends critically on
ascertaining the correct overall uncertainty.
Rarely is what you measure directly comparable to other experimental results or theoretical calculations; typically, you must process your data
0.2. ERROR ESTIMATION AND PROPAGATION
3
through various formulas to extract a parameter you can compare with others. Therefore, it is critically important to accurately propagate the uncertainties in the original data through the calculations and arrive at a reasonable uncertainty for the nal value. This is the process of error propagation.
These are the issues that we now discuss.
0.2.1
Denition of Uncertainty
The rst step involved in error estimation is to identify the possible types of
errors that can occur in your experiment. There are three basic types you
need to be aware of: illegitimate errors, systematic errors, and random errors. Illegitimate errors are faults in experimental procedures or calculational
blunders. We will make every eort to avoid making these kinds of errors,
but if we do blunder, we can easily nd and correct them; we will assume
that we have eliminated all illegitimate errors from our experiments. This is a
formidable assumption since the time to perform our experiments is limited;
however, the procedures are not all that complicated; so this assumption
should not be a bad one. Under this assumption therefore, we cannot use
illegitimate errors as reasonable explanations for any discrepancies that ultimately occur in our analysis. To discuss the other types of errors, we must
more carefully distinguish between accuracy and precision.
Accuracy represents how close a measurement is to the true value. Precision indicates how well the results of an experiment have been determined,
independently of how well the results agree with the true value. This tells
us about the self-consistency of a measurement. When judging the results
of an experiment, we must consider both the accuracy and the precision. In
general, when we quote the uncertainty of an experimental result, we are
referring to the precision with which the result has been determined.
Systematic errors are errors that make our results dierent from the true
value in a reproducible way. They are usually due to the faulty calibration
of equipment or some unknown bias on the part of the experimenter. They
can be subtle and hard to quantify. Knowledge of the apparatus and the
experimental procedure is the central manner of minimizing the impact of
systematic errors in our results. Such errors aect the accuracy of our results,
since they contribute the same amount of discrepancy each time we perform
the experiment. Random errors, on the other hand, constitute the major
source of imprecision in an experiment. These are the random uctuations
in measurements from experiment to experiment, primarily due to the nite
4
CHAPTER 0. INTRODUCTION
resolution of our apparatus. To control random errors, we must perform the
experiment many times and use a statistical analysis to extract our results.
A given accuracy implies at least an equivalent precision; thus, accuracy
depends on these uctuations too.
To clarify the dierence between these two types of errors, consider the
simple experiment of determining the average speed of a rolling ball as it
passes by a meter stick, by using a handheld timer. A systematic error
involved in this experiment could be due to the calibration of our instruments.
For instance, assume that the intervals on the meter stick are 1% larger
than they should. Then, every time we record a distance of 50 cm the
actual distance travelled by the ball is 50.5 cm. The distinct feature of
systematic errors, such as this, is their repeatability. No matter how many
times we perform the same measurement, if we are using the same instrument,
we will always make the same mistake. Random errors dont share this
property. In the rolling ball experiment a random error is due to the timer
operator. No matter how hard he or she tries, a human operator cannot
be entirely consistent on when to press the start/stop button on the timer.
Sometimes, he or she will start the timer just a little before the ball passes
by the predetermined mark on the ruler, some times a little after. The
same thing will happen when it is time to stop the timer. The end result
is that, if we perform many attempts at the experiment (and assuming that
the ball is always launched at the same speed,) the times we will get will
vary randomly around the correct value. A random error is also introduced
by the nite resolution of the devices we are using. Suppose that we have
replaced the timers human operator with a perfect photogate setup. If
the smallest time increment on our timer is 0.01 s and we measure a time
of 50.23 s then we only know that the actual time is between 50.225 s and
50.235 s. Every value in between these two will be rounded and displayed on
our instrument as 50.23 s. If we perform the measurement many times and
we always get the same reading of 50.23 s, the reasonable thing to assume is
that, in every attempt, the actual time was dierent, but within the abovementioned limits.
We also distinguish between what we call absolute and relative uncertainty. Absolute uncertainty is the uncertainty in a quantity expressed in the
same units as the quantity. For example, if we write
g = 9.7990 0.0014 m/s2 ,
then the absolute uncertainty in the measurement of g is g = 0.0014 m/s2 .
0.2. ERROR ESTIMATION AND PROPAGATION
5
Note that we always consider the absolute uncertainty to be a positive number. The relative uncertainty is the uncertainty expressed as a fraction or
percentage of the quantity. The relative uncertainty of g in our example is
g
0.0014 m/s2
= 0.00014 = 0.014%.
=
g
9.7990 m/s2
0.2.2
Estimating Parameters and Their Uncertainties
Now that we know what constitutes error, we should describe how to estimate
it given a data set. Lets consider the most common situation in which
we need to extract an estimate for the error; this occurs when weve made
several measurements of the same quantity and we want to extract an average
value and a corresponding uncertainty. For concreteness, suppose we have
a set of N data points xi with uncertainty xi which we do not assume is
the same for each of the measurements. (We need this generality because
the uncertainty can vary independently of the quantity being measured; for
example, if you measure a current value on two dierent scales on a meter,
then the uncertainties of the two measurements are dierent.) Then you can
show (c.f. Bevington and Robinson, see the Preface for the reference) that,
assuming a Gaussian error distribution, the most probable value is the
mean, or average, value, x:
x=
N
xi
i=1 (xi )2
N
1
i=1 (xi )2
.
(0.1)
This formula may look complicated, but all it does is give prominence to
those measurements with the smallest uncertainty, a very reasonable thing
to do. Notice that, if all the values have the same uncertainty, xi = x
for i = 1 to N , this formula reduces to the usual x = (1/N ) N xi . The
i=1
corresponding uncertainty in the average value is the standard deviation,
, and comes from the relation
1
=
2
N
1
.
2
i=1 (xi )
(0.2)
We can see from this formula that the uncertainty of the combination is always less than the smallest uncertainty of the component measurements. We
6
CHAPTER 0. INTRODUCTION
should certainly hope so! After all, the more measurements we do, the better
we should know what the answer is. In the case that all the uncertainties
are equal to x, this expression reduces to = x/ N , showing that the
uncertainty decreases at the relatively slow rate of one over the square root
of the number of measurements.
Suppose you have three measurements of the resistance of a resistor that
have come from three dierent techniques. This data appears in Table 0.1.
The nominal value is the value printed on the resistor; the multimeter measurement comes from an ohmmeter measurement; and the current and voltage measurements come from a detailed analysis using Ohms law. Each
technique has its own associated uncertainty. We want to combine these separate measurements into a single estimate of the resistance. Since the errors
are distributed symmetrically (i.e., another measurements value is equally
likely to fall on either side of the current measurements value), we can reliably use our formulas based on a Gaussian distribution, i.e., the ones given
above. We nd the resistance to be
2.5866 0.0027 k.
This answer reects several general features of these formulas mentioned
above. First, it is closest to the measurement with the smallest uncertainty.
Second, the uncertainty of the combined measurements is less than the smallest measured uncertainty, but not by much. This reects the fact that the
other measurements uncertainties were much larger than the last. This is
good intuition that you should incorporate into your thinking patterns; it
will help you identify the important sources of error in your experiments.
Experimental Technique
Nominal Value
Multimeter
Current and Voltage Measurements
Value
2.70 0.14 k
2.69 0.01 k
2.5785 0.0028 k
Table 0.1: Resistance values for a single resistor from dierent measurements.
For more sophisticated experiments, we dont necessarily measure the
same quantity over and over. We might change one parameter and measure
another, to investigate the eects of one on the other. When we do this,
we are looking for correlations between the dierent parameters. One of the
0.2. ERROR ESTIMATION AND PROPAGATION
7
most eective ways to spot correlations is to graph the parameters and look
for some functional relationship. The easiest and most reliable functional
relationship to recognize and quantify is a linear one, i.e., when you plot the
parameters, the data points fall along a line. In fact, this type of correlation
is so important that we sometimes alter the parameters that we plot to force
the graphed data into a line, as in a log-log plot or a semi-log plot. Here, we
dont just plot the data, but certain functions of the data that are linearly
related. This cant always be done, but for our labs it can; so, we will focus
solely on analyzing linear correlations. In this case, the correlation between
the two quantities is described in terms of two numbers: the slope of the line
and its y -intercept. For most purposes, the slope is the more important of
the two, but the intercept can also contain important physical information.
Once the data suggests a linear relationship, we want to extract the slope
and intercept of the best t line.
There are many complicated denitions of best t that one can use
to extract a consistent slope and intercept from a set of data. The most
often used method goes by the name of least squares t. The reason for the
popularity of this particular method has to do with its relative simplicity and
statistical signicance. It will provide the slope and intercept of the most
probable line to t a set of data, assuming a Gaussian distribution for the
errors. Thus you can think of this as analogous to extracting the average
quantity for a larger class of measurements.
We can derive the formulas for the linear least squares t to a set of data
with the following assumptions: rst, that the uncertainty is symmetrically
distributed about the data values, and second, the uncertainty in the dependent variable is more signicant than the uncertainty in the independent variable. These assumptions are discussed at length in Bevington and Robinson,
and we will simply take them for granted. With these assumptions in mind,
we can proceed as follows: given N data points (xi , yi yi ), i = 1, . . . , N ,
we want to determine the parameters of the line y = ax + b so that the square
of the vertical distance between the y coordinates of the line and the data,
weighted by the uncertainty yi and summed over all the data points, is a
minimum. The geometry of this construction is shown in Figure 0.1. That
is, we want to minimize the function of 2 variables
N
e(a, b) =
i=1
1
(yi axi b)
yi
2
To minimize this function, we take the partial derivatives with respect to
8
CHAPTER 0. INTRODUCTION
y
y=ax+b
Minimize the
squares of the
deviations
x
Errorbars have been suppressed
to avoid ambiguity.
Figure 0.1: The least squares method determines the line that minimizes the
square of the vertical distances between the line and the data.
a and b and set them equal to zero. This yields the following system of linear
equations for a and b:
N
a
N
x2
xi
i
+b
=
2
2
i=1 (yi )
i=1 (yi )
N
1
xi
+b
=
a
2
2
i=1 (yi )
i=1 (yi )
N
N
xi yi
2
i=1 (yi )
N
yi
2
i=1 (yi )
Using your favorite technique, you can show that the solution of this linear
system of equations is
a=
1
D
b=
1
D
N
1
2
i=1 (yi )
N
x2
i
(yi )2
i=1
N
xi yi
2
i=1 (yi )
N
N
yi
2
i=1 (yi )
xi
2
i=1 (yi (yi )
N
xi
2
i=1 )
N
yi
2
i=1 (yi )
N
1
D=
2
i=1 (yi )
N
x2
i
(yi )2
i=1
N
N
xi yi
2
i=1 (yi )
where
xi
2
i=1 (yi )
,
(0.3)
2
.
(0.4)
0.2. ERROR ESTIMATION AND PROPAGATION
9
The corresponding uncertainties in the t parameters are
(a)2 =
1
D
(b)2 =
1
D
N
1
,
2
i=1 (yi )
N
x2
i
.
(yi )2
i=1
(0.5)
Lets take a look at these equations in action. Consider the data given in
Table 0.2; this data came from current and voltage measurements across a
resistor of unknown resistance. Ohms law indicates that, for most resistors,
the voltage is linearly related to the current, the proportionality constant
being the resistance. Weve plotted this data in Figure 0.2, which strongly
suggests a linear relationship between current and voltage. So, it makes sense
to apply our least squares equations directly to the current-voltage data.
Considering the data again, we see that the errors appear symmetrically
distributed around each point, and the independent variables (the currents)
uncertainty is much smaller than the corresponding uncertainty in the voltage
data. So, this data satises both conditions for applying the least squares
analysis.
Current
(mA)
0.1936 0.0001
0.289 0.001
0.388 0.001
0.575 0.001
0.946 0.001
1.042 0.001
1.144 0.001
1.196 0.001
1.484 0.001
1.750 0.001
Voltage
(V)
0.719 0.001
0.813 0.001
1.093 0.001
1.620 0.001
2.66 0.01
2.93 0.01
3.22 0.01
3.37 0.01
4.17 0.01
4.93 0.01
Table 0.2: Current and voltage data for computing the resistance of a resistor
using Ohms law.
We nd (and you should verify with units!) the following values for the
10
CHAPTER 0. INTRODUCTION
sums involved:
10
xi
= 1.5212 106
(yi )2
i=1
10
yi
= 4.4578 106
(yi )2
i=1
10
x2
i
= 7.0201 105
(yi )2
i=1
10
xi yi
= 2.0107 106
(yi )2
i=1
10
1
= 4.0600 106
(yi )2
i=1
from which we nd D = 5.3607 1011 and then
a = 2.5785 0.0028 k
b = 0.1319 0.0011 V
The line with this slope and intercept is the line drawn in Figure 0.2. We
see that this line is a very good representation of the data as it goes through
most, but not all, of the data points within error. The equation of this line
is y = a x + b or since our y is voltage and our x is current: V = a I + b.
We will do a lot of such plotting in the labs using the program KaleidaGraph to do the work. It will return the slope and intercept of the line, but
it is critical to understand exactly what the weighted least squares line t
is actually telling us. That is, we have a little more work to do. Working
through the units, the slope is in k and the intercept is in V. It seems, then,
that the slope is the resistance of the circuit, but does this make sense in
light of the theory that we have?
The next few steps are simple, yet somehow subtle for students to comprehend about the relationship between the slope of a weighted least squares
(WLS) t and the physical quantities of interest, so please follow carefully,
making sure you understand each step.
11
0.2. ERROR ESTIMATION AND PROPAGATION
5.0
4.5
4.0
V(V)
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.0
0.5
1.0
I(mA)
1.5
2.0
Figure 0.2: Plot of voltage versus current for the data in Table 0.2.
1. Write down the equation of the WLS t line: y = a x + b
2. Substitute the dependent and independent variables actually plotted
for y and x, respectively. For our plot: V = a I + b.
3. Write down the theory that describes the data taken. In this case,
Ohms law says: V = I R.
4. What we want to do is make this equation the same as the equation for
the WLS t line so that we can compare the two. Using basic algebra,
we can rearrange Ohms Law to V = (R) I without loss of generality.
5. Now this last equation is starting to look like the equation of the WLS
line. It has the dependent variable, V, equal to the independent variable, I, times what we assume to be constant, R. This all looks good,
but it has no y-intercept. However, we are free to add zero to any equation so lets add zero to the right hand side, giving: V = (R) I + 0.
6. All that is left is to compare the two equations. To this more clearly,
12
CHAPTER 0. INTRODUCTION
lets write them down one on top of the other:
V = (a) I + b
V = (R ) I + 0
7. Note that our theory, Ohms Law, predicts that these two equations
are equivalent, so the WLS t of our data provides a test of the theory.
(This is the whole point of plotting the data in the rst place.) The
only way that this equality can be true is if the coecients of the independent variable, I, in both equations are equal, and the two constants,
b and 0, are also equal. In other words: a = R and b = 0.
8. In summary, the prediction made by Ohms Law is that a plot of Voltage
vs. Current will have a slope equal to the resistance and an intercept
equal to zero, thereby allowing us to interpret the slope and intercept
data returned by the WLS algorithm in terms of physical quantities.
9. We will have more complicated theories, but the process is ALWAYS
THE SAME. We derive a linear relationship between the measured
quantities and plot them to test the theory and to derive other physical
quantities. Sometimes, the intercept wont be predicted to be zero or
the relationship between the slope and the physical quantities requires
additional algebra, but it only involves straightforward manipulation
of the equations. If it is done step-by-step using this method, it will
work every time.
Lets analyze the results a little. Specically, the intercept has an interesting interpretation here. It is not zero within experimental uncertainty.
This means that this data does not completely support the theory. Whereas
if it were, the data would support Ohms Law completely. Since it also means
that with zero current there would still be a voltage across the resistor, we
should think about what may have happened to cause this apparent contradiction of Ohms law. It may be bad data or a limitation of the theory.
0.2.3
Propagating and Reporting Uncertainties
At this point, you should have a clear idea of what uncertainty is and how
to estimate it in some simple cases. Once you have your estimates of the
parameters of interest and their uncertainties, you will likely want to run
0.2. ERROR ESTIMATION AND PROPAGATION
13
them through some formulas to arrive at numbers you can compare to other
peoples measurements. This brings us to discuss the propagation of uncertainties through functions and formulas.
To keep things simple, we will make the assumption that the uncertainties
in your parameters are symmetrically distributed about the average and that
the parameters are independent of each other. That is, two measurements of
dierent parameters are uncorrelated. This is not always true; for example,
in an ideal gas at xed pressure, the density and temperature uctuations are
linked by the equation of state. However, the added complications needed
to account for these eects are typically intimately tied to the physical system youre studying, making a general treatment cumbersome. By ignoring
correlations and assuming symmetry, we can reduce all the necessary error
propagation down to some simple calculus.
Suppose we have a parameter with its uncertainty: x x. The question
we want to answer is What is the uncertainty of some function, f , of this
data? Under our assumptions, the answer comes from the Taylor series
expansion of f (c.f. Bevington and Robinson): f (x + x) = f (x)+ f (x)x +
O (x2 ). From this we nd the uncertainty f in the function value f (x) is
f =
df
x ,
dx
with the derivative evaluated at the point x. We can generalize this result to
functions of several variables as follows: given the data x x, y y, . . .,
the function f (x, y, . . .) has the associated uncertainty
f =
f
f
x +
y + . . . ,
x
y
where all the derivatives are evaluated at the point x, y, . . .. If we recall that
we dened absolute uncertainties to be positive, we can write this as
f =
f
f
x +
y + . . . ,
x
y
(0.6)
From this relationship, we can derive all the familiar results of error propagation.
14
CHAPTER 0. INTRODUCTION
Example: Addition and Subtraction
Given: f (x, y ) = 3x + y z + 5
Find: f
f
f
f
x +
y +
z
x
y
z
= |3| x + |1| y + |1| z
= 3x + y + z
f =
Example: Multiplication and Division
Given: f (x, y ) = x2 y/(5z )
Find: f
f =
f
f
f
x +
y +
z
x
y
z
= |2xy/(5z )| x + x2 /(5z ) y + x2 y/(5z 2 ) z
Example: Ohms Law
Given: V = IR
Find: R
R =
=
R
V +
V
1
(V ) +
I
R
I
I
V
I
I2
Now that we have a clear idea of what constitutes the uncertainty of a
measurement, how to estimate it, and how to propagate it, we should talk
about the proper way to report the uncertainty of a measurement. This
forms the subject of signicant gures. Here is how you should determine
the number of signicant gures:
1. Calculate the uncertainty in the quantity.
2. Round o the uncertainty to one or two digits.
3. Express the uncertainty in the same units as the quantity measured.
0.2. ERROR ESTIMATION AND PROPAGATION
15
4. Round o the quantity to the last decimal place of the uncertainty.
5. Always write down the nal result of a calculation with the uncertainty
and the units included.
Use the form
(2.34 0.23) 103 m, or 2.34 0.23 km,
not expressions such as
2.34 103 m 0.23 103 m,
2.34 km 23 101 m,
2340 m 0.23 103 m.
These are the rules you will use most often in reporting your results. They
become rather cumbersome, though, when you begin to make very precise
measurements. Consider, for example, the charge on the electron; the best
measurement we have of this number is
(1.60217733 0.00000049) 1019 C.
This is very annoying; so, weve developed a shorthand for reporting these
kinds of measurements. You simply quote the result to the known uncertainty and place the uncertainty of the last few digits in parentheses after
the number and before the power of ten. In this notation, the electrons
charge is
1.60217733(49) 1019 C,
which is much easier to deal with. If you begin to make measurements of
such precision that you need to employ this convention, feel free to do so.
Finally, in various experiments we quote what are called accepted values
for various physical parameters. These are the scientic communitys best
estimates of these numbers. They have been experimentally veried and
checked for consistency with other measurements. Most you will nd are
very precise, typically 6 or 7 decimal places. You will discover in trying
to do your own labs that making such high precision measurements is not
easy. They also let you know that there is still some uncertainty in these
parameters; they are not exact; but you will probably not be able to help
narrow that using the equipment and techniques we have, which means they
are exact as far as we can tell. So, keep in mind as you attempt to verify
these numbers, that other folks had to do these measurements too.
16
CHAPTER 0. INTRODUCTION
0.3
The Lab Worksheet
The lab worksheets are formatted to be well organized, concise, and complete,
and designed to maximize eciency. The worksheets are organized as follows:
Purpose
Procedure
Computer Work
Pre-Classroom Checklist
Calculations & Analysis
Discussion and Conclusion
Each item is described in detail below. Some may occur more than once in
any given worksheet.
0.3.1
Purpose
The Purpose section will give a general idea of the theories being tested and
the experiments to be performed. However, this is expected to be a reminder
of what has been thoroughly studied in preparing for the lab, and will not
be sucient to answer the questions in any detail.
0.3.2
Procedure
The Procedure section provides a guide to the experimental set-up, data
collection, and necessary calculations. This section will always be the second section of the worksheet, but there may be a few of them in any one
worksheet, one for each new experiment in that days lab. Since 103N is an
electromagnetism and optics lab, you will be connecting circuits and manipulating optical equipment such as lasers. Figures are provided and numbered
0.3. THE LAB WORKSHEET
17
to show how to set up connections and place equipment correctly. The gures are your friends in setting up the experiments. Do not ignore them.
They provide critical information not found in the text.
After setting up the experiment, follow the directions given to begin data
collecting, sometimes one piece of data, other times twenty. Any data collecting will be specied and organized by a table or space. Blank spaces above
answers are to show your work in reaching the answer entered below the
space. This is very important as indicated as no credit is given for
answers without work.
When recording data, be that in a table or on a line, present your raw
data neatly and completely including units, uncertainties, and signicant
digits. Any calculations used in recording the data should be shown in the
space provided above the answer. Show all calculations keeping numbers out
of the calculations until the nal step. An example of how to do calculations
properly will be given in the Calculations & Analysis section that follows.
When showing work, it is critical to propagate all units through the calculations, and keep the work organized. This convinces the reader of the
validity of the calculations. Do not ignore units for several steps of a computation and then just write down what seems to be the proper ones at the end.
It is very easy to be o by several orders of magnitude (e.g. using k instead
of M) by not carrying units through the calculation, and nearly impossible
to track down the source of the error. Remember the time pressure of this
lab and that there is no time to waste nding these types of mistakes.
Before moving on to the next section of the worksheet, double check that
all required work is completed. Remember that the procedure section is not
only for data recording, but also has short answer questions and calculations.
Complete everything in each section before moving on to the next one. Lab
partners can divide the work to be more ecient but make sure everything
is nished.
0.3.3
Computer Work
After collecting and recording data you will usually make a graph, but think
about why the data is being plotted. The graphs will give vital information
used in the calculations, analysis, discussion and conclusion. Graphing is
typically the easiest and most accurate way to get the information. Graphing
is done on the computers in lab, meaning they must be done in the two hour
lab time and will be handed in at the end of the third hour with the worksheet.
18
CHAPTER 0. INTRODUCTION
KaleidaGraph is the plotting program you will be using for almost all
of your graphs. The rst two lab sections focus on learning and practicing
KaleidaGraph skills. Most labs require graphing data sets and making linear
ts to them. Therefore, you need to become procient in KaleidaGraph to
complete the labs on time. Title graphs appropriately, include all units, and
label axes. Despite the fact that the computer will be doing most of the
work in graphing data, it is important to understand what the computer is
actually doing. To further this understanding, the second lab 0.W2 requires
graphing by hand. The next sections describes how to do this.
Hand-Fit Graphing
Use good quality graph paper with a resolution of at least 4 grids per inch.
Each graph should be large and clear; a full page graph of 10 points is not
unreasonable. The scales on the axes should be appropriate for the data
ranges, i.e. so that the data covers most of the graph. Having bunched
up data points leads to diculty in reading the graph and loss of precision
in tting lines and calculating slopes and intercepts. Do not draw your
axes across a full page and choose your scale in such a way that the data
points occupy only a few cm2 ! Also, understand the dierence between the
dependent and independent variables when graphing. The quantity which is
the function of the other in the experiment is conventionally plotted along
the vertical axis. If asked to plot y vs. x, for example, interpret y is the
dependent variable and plot it on the vertical axis. Always label the axes by
the physical quantity plotted and include the correct units in parentheses.
As with computer plotting, when tting data by hand do not connect the
dots. Doing so has no physical basis and obscures insight into the physics.
Instead, using a straight edge, eyeball a line that goes through the error
bar of every data point and also minimizes the square of the distance to all
data points. Remember that the line represents the trend present in your
data and might not pass through any data points. The line drawn gives the
best t values for the slope and intercept. To calculate the error in the slope
and intercept, draw the steepest and shallowest lines that are consistent with
both the trend of the data and stay within the error bars. The uncertainty
in the t parameters are given by the formulae
a =
|asteep ashallow |
2
19
0.3. THE LAB WORKSHEET
b =
|bsteep bshallow |
2
.
When calculating slopes for all of these lines, choose two convenient points
on the lines that may or may not be data points. The t line is far more
important than individual data points because it is a type of averaging of
the values of the data set.
Although the estimate the best t line by hand is somewhat subjective,
the computer is just doing a more sophisticated and reproducible version of
this. But the hand-tting exercise gives an intuition of what the computer
is doing and how to estimate these things independently as a check on the
results from the computer. Blind faith in a computer program has hurt many
a researcher.
0.3.4
Pre-Classroom Checklist
The checklist is included to help ensure that all in-lab work is completed
before leaving for the classroom. There are only 2 hours in the lab to complete
the procedures and graphs. Another class enters the lab when your class
leaves so there is no opportunity to go back and use the computers or redo
a portion of the experiment. The Lab Worksheets have been designed to
be completed in parts. Each procedure and its computer work is completed
before moving on to the next experiment. In this manner, you will be able
to do a complete analysis and discussion on at least some parts even if you
dont nish the lab.
There are
provided to physically check o. Do this. It will keep you
organized when you get to the classroom. One hour is not a lot of time to
nish all the calculations, discussions, and conclusion. Also make sure that
each partner has her/his own data, graphs, tracings, etc.. Contact amongst
lab partners is not allowed in the classroom.
0.3.5
Calculations & Analysis
At this point, all Procedures and Computer Work is nished. All the circles
in the check list are checked to ensure that all the laboratory work is nished.
Once the two hour lab period is over, the class moves to the classroom and
nishes the worksheet in one hour. However, if lab partners complete all
lab sections sooner than two hours they may begin the in-classroom sections.
20
CHAPTER 0. INTRODUCTION
The necessary calculations are clearly stated with referenced equations,
gures, and tables when appropriate. There is a line or space left for each
answer. There is also room to show your work. You must show all the
steps and reasoning behind your answers to get full credit. A sample calculation is done here as a future guide. Suppose the question in the
in-classroom calculation & analysis section reads:
Calculate the resistance in a circuit given the voltage, V = 5.00 0.01 V ,
and current, I = 20.0 0.2 mA, with uncertainties and units. Show Work.
V
R
R
R
R
R
R
R
= IR
V
=
I
5V
=
20 mA
= 250
V
1
(V ) + 2 I
=
I
I
5 V
1
0.01 V +
0.2 mA
=
20 mA
400 (mA)2
= 3
= 250 3
The above is what to show and how to report the answer in proper form.
Note that numbers were This same calculation is repeated later in the lab.
The Calculation & Analysis sections can also ask questions about your
understanding of the physical principles behind the lab. Answer all the questions completely, stating your reasoning and show any calculations or drawings necessary to demonstrate your knowledge. You should now understand
the aspects of the lab well enough to write the concluding statement.
0.3.6
Discussion and Conclusion
The nal section brings together everything learned in the lab to answer
questions and make specic, concrete conclusions about the physics studied.
Specically provide:
A sentence or two describing the purpose of the experiment, i.e. the
main theories, predictions, important formulae, etc.
0.3. THE LAB WORKSHEET
21
State your most important experimental values including units and
uncertainty and indicate whether these results agree within uncertainty
with accepted values and/or theory.
Candidly address the uncertainties in the lab and attempt to unambiguously and uniquely identify the key sources of error. This is especially
true if your experimental results do not agree with accepted values
or theory. Remember: This discussion should not include things like
human error or errors in the calculations. These illegitimate errors
can and should be absent from our lab work.
With all these ideas clearly laid out, state whether the experiment was
a success or not. This primarily means describing the degree to which
the experimental results support the theory or theories being tested.
Note: NEVER claim that the experimental values or the entire lab agreed
with theory when it did not. Being disingenuous in answering these questions
will not only result in a low score and an irritated grader, but also, and most
importantly, it is bad science. Sometimes (even frequently) an experiment
doesnt work properly and it is important is to analyze why, to learn from the
mistakes made. Finally, the instructor may also assign additional questions
to ponder and to incorporate into the above discussion. Typically, these
questions have direct answers, but only after thinking about the lab.
22
CHAPTER 0. INTRODUCTION
0.W1
Error Analysis Worksheet
0.W1.1
Purpose
The purpose of this rst lab is to become familiar with plotting data in
Kaleidagraph and propagating error correctly.
0.W1.2
Procedure
Perform the KaleidaGraph practice plot and analysis described below in the
Computer Work section, and attach it to the worksheet that follows.
0.W1.3
Computer Work
Using KaleidaGraph for Data Analysis
The software package known as KaleidaGraph is a useful tool for data analysis. Of course, it will only be useful after developing some skill with the
software. It is not dicult to use, but like all software, takes a little time to
learn. This lab is not meant to be a comprehensive guide to using KaleidaGraph, but it will demonstrate how the most useful features of KaleidaGraph:
plotting and tting curves to data.
A Model Experiment.
Suppose an experiment has been performed to test a theory that predicts the
magnetic eld strength of a material is inversely proportional to its temperature. This theory can then be represented by the equation:
1
B=a ,
T
where B represents the eld strength, T is the temperature of the material,
and a is a proportionality constant. By plotting the magnetic eld strength
against the reciprocal of the temperature and applying a WLS t, we expect
a straight line with a slope equal to the proportionality constant, but if and
only if, the stated theory is correct.
Note that the proposed linear relationship is not between B and T, but
between B and 1/T, so we have had to adjust what we are plotting to apply
a linear t. Another example is the distance versus time relationship used to
0.W1. ERROR ANALYSIS WORKSHEET
23
describe acceleration of a falling object d = (1/2)at2 . We can apply a linear
t to the graph by plotting distance versus t2 as will be shown next week.
Using a linear t in this way provides a scientic test of the theory because
if the t is poor, the theory is not well supported by the data, but if the linear
t is very good, the data tends to support the theory. Using linear ts of
measured data will be repeated throughout the semester and is a valuable
tool in scientic research, but lets get back to the temperature dependence
of magnetism.
The following table lists the magnetic eld strength, B, in milli-Gauss,
taken with a magnometer and the temperature of the material, T, in degrees
Kelvin, taken with a thermometer.
Magnetic Field Strength vs. Temperature
B (mG)
T (K)
100 1
0.43 0.01
250 1
0.15 0.01
500 2
0.09 0.01
750 2
0.06 0.01
1000 3
0.04 0.01
0.W1.4
Enter Data
Start KaleidaGraph and notice that it launches a data window. Its
default name is Data 1.
Activate the data window by clicking on it.
We will enter all the magnetic eld data in column A, their uncertainties in column B, the values for temperature in column C and
their uncertainties in column D.
Click the rst data cell and begin with the rst magnetic eld value.
Move to other cells using the mouse, arrow, Tab or Return keys, and
enter the entire data table.
Note that KaleidaGraph plots data in a column vs. column fashion, so
the x and y-coordinates of a single data pair should be placed in the
same row, but in dierent columns.
24
CHAPTER 0. INTRODUCTION
Renaming Columns of Data
This might not seem important at rst, but KaleidaGraph labels the axes
on plots with the name of the columns, so renaming them is essential. The
default names of columns are A, B, C, etc. These appear in the
column title row of the data window, along with a number. To change the
the names:
1. Double-click on the column title.
2. In the Column Format: dialog box which appears there will be a list
of column titles. Highlight the title by clicking on it.
3. Type in the new title of the column (for example A becomes B (mG)).
4. Make sure to name each column including the uncertainty columns.
For example, Column B becomes dB (mG). Also indicate the units
in each column.
5. When nished changing names, click on the button labeled Done.
0.W1.5
Entering Formulas
Often, the raw data entered is not immediately in the form needed for plotting. Never fear, KaleidaGraph is capable of performing mathematical operations on the data by dening a formula. Formulas tell KaleidaGraph to put
in one column the result of operations on data in other columns.
In KaleidaGraph, formulas refer to column numbers rather than the
column names. The zero or rst column may be any column and can be set
by merely clicking on the column title cell and changing it to column zero.
The columns to the right then take the numbers 1, 2, 3.... The columns to
the left become unnumbered.
The syntax of formulas is
cx = f(cy , cz , ...)
where x, y, and z are the numbers of the columns which contain the data to
be operated on, and f(...) is the mathematical expression KaleidaGraph will
calculate.
0.W1. ERROR ANALYSIS WORKSHEET
25
For example, to plot B vs. 1/T , a column containing the reciprocal
values of the temperature is necessary. Make one by entering and executing
a formula using the following steps:
1. From the Windows menu at the top of the screen, select the option
Formula Entry.
2. In the Formula Entry window which appears, click on one of the
buttons labeled F1 - F8.
3. Type in the formula in the space provided in the Formula Entry
window (in this case, c4 = 1/c0).
4. Click the button marked Run in the Formula Entry window.
The button labeled F1-F8 corresponds to one of the function keys at the
top of the keyboard. Pressing that F-key will bring up the Formula Entry
window again, and it will still contain the formula, thus saving important
formulas for later use. During the semester, we will need to create some
complicated formulas so use a NEW F-key for each formula. That way the
old formula is not overwritten.
Make sure to rename the new 1/T column calculated with the formula
entry so it is clear what that column contains. Finally, the uncertainties
in 1/R are dierent from the uncertainties in T so use KaleidaGraph to
also calculate an uncertainty column for this data. Determine the necessary
formula by using error propagation as described in the manual, and rename
this column, too.
0.W1.6
Plotting Data
Making a Scatter Plot
Because our data will vary about the best t line, we never want to connectthe-dots when plotting. Use a Scatter plot instead:
1. Activate the window containing the data by clicking on it.
2. From the Gallery menu at the top of the screen, select the Linear
submenu, and from that select the Scatter option. (If instead the
Line option is selected, KaleidaGraph will connects the dots, which is
what we do NOT want.)
26
CHAPTER 0. INTRODUCTION
3. A dialog box will appear. In it, there will be columns of circles labeled
X and Y. Under X click on the circle in the row containing the
title of the column which contains the x-coordinates of the data. A
solid black circle should appear.
4. Do the same with the y-coordinates in the column labeled Y.
5. Click on the button labeled New Plot.
Adding Error Bars
Note that there are no error bars so they must be added.
1. Activate the window containing the plot by clicking on it.
2. From the Plot menu at the top of the screen, select the option Error
bars...
3. In the Error Bar Variables dialog box which appears, click on the
square labeled X Err.
4. Click and hold on one of the two rectangles labeled % of values, and
select the option Data Column.
5. Select the column which contains the uncertainties in the x-coordinates
of your plotted data.
6. Click on button labeled OK.
7. Now, follow the same procedure starting with the square labeled Y
Err.
8. Click on button labeled Plot.
0.W1.7
Performing a Weighted Least-Squares Fit
We now come to the moment of truth and will apply the linear t to the
plot. Both the slope and y-intercept of a linear t are usually important
pieces of information, but they are meaningless without uncertainties. One
of the great benets of KaleidaGraph is that it will calculate them if the t
is done properly, so use the following steps:
0.W1. ERROR ANALYSIS WORKSHEET
27
1. Activate the window containing the plot by clicking on it.
2. From the Curve Fit Menu at the top of the screen, select the General
submenu, and from that, select the option t1.
3. In the Curve Fit Selections: dialog box which appears, click on the
button labeled Dene...
4. In the new dialog box that appears, click on the square labeled Weight
Data so that an X appears in it.
5. Click on the button labeled OK.
6. In the Curve Fit Selections: dialog box, click on the square next to
the column title which contains the error in the y-coordinate of the
data.
7. A new dialog box will appear called Weight Data From Column:.
By clicking on the
and
buttons, select the name of the column
containing the uncertainties for the y-coordinates.
8. Click on the button labeled OK.
9. Now click on this button windows labeled OK.
To display the numerical results of the t (the slope and y-intercept with
uncertainty), simply choose the Display Equations option from the Plot
menu, and a table containing the results will appear. Note that, in this table,
m1 is the y-intercept and m2 is the slope of the best t line.
The Work You Should Turn In
After following the above example, attach the printout of the plot to the
worksheet that follows. This printout should contain: a descriptive title,
properly labeled axes, x and y error bars, and the best t line plotted by
KaleidaGraph. Also, at the bottom of the page below the plot report in a
complete sentence the value for the constant of proportionality with units
and uncertainty!
The TA may require more so follow her or his instructions.
28
CHAPTER 0. INTRODUCTION
0.W1.
IN-CLASSROOM CALCULATIONS
0.W1
Name:
29
In-Classroom Calculations
Day/Time:
Instructions: Perform all of the following calculations using the techniques
explained in Chapter 0 (Introduction) of the lab manual. Show all calculations explicitly and propagate uncertainties where appropriate. Write all
answers in proper form including the correct number of signicant gures
and units, e.g. x = 1.013 .021 m.
1. Four independent measurements of the voltage supplied by a certain
D-cell battery were made:
2.4 0.6 V
2.96 0.08 V
3.02 0.06 V
2.968 0.004 V.
Referring to 0.2.2, calculate the most probable value of the D-cell
voltage as well as the standard deviation of the measurements using
equations (0.1) and (0.2).
30
CHAPTER 0. INTRODUCTION
2. Refer to 0.2.3 for the calculation and propagation of uncertainty.
Two lengths have been measured to be L1 = 4.8 1.2 cm and L2 =
3.2 1.6 cm.
(a) Calculate the sum L = L1 + L2 and its absolute uncertainty, L.
Use these to calculate the relative uncertainty in L.
(b) Calculate the dierence L0 = L1 L2 , as well as its absolute and
relative uncertainties.
Compare these uncertainties with those in the sum.
(c) Now calculate the product P = L1 L2 and its absolute and relative
uncertainties.
0.W1.
IN-CLASSROOM CALCULATIONS
31
(d) Calculate the quotient Q = L1 /L2 , its absolute and relative uncertainties.
Compare the quotient uncertainties to those in the product.
3. The area of a square has been measured to be A = 50 6 cm2 . What
is the length of one side of the square? Remember to derive formulas
for the value and error before calculating anything.
32
CHAPTER 0. INTRODUCTION
4. Two resistors, with resistances R1 = 540 54 and R2 = 860 86 ,
are connected in parallel. Calculate the equivalent resistance, Req , of
the combination using the formula. Hint: Solve for Req rst.
1
1
1
=
+
.
Req
R1 R2
5. For = 60 3 , calculate sin and tan . Hint: Convert the angle to
radians.
0.W1.
33
IN-CLASSROOM CALCULATIONS
6. Given that L = 20 4 cm and y = 8 2 cm in the triangle
y
L
calculate sin . Hint: Use trigonometry to express this function in terms
of the length of the triangle legs and/or hypotenuse.
Attach your KaleidaGraph plot to the end of this worksheet.
Bring graph paper next week.
End Error Analysis Worksheet
34
CHAPTER 0. INTRODUCTION