20
08
8,
PC237 Lab Manual
ru
ar
y
Terry Sturtevant1
Fe
b
February 8, 2008
1 Physics
Lab Supervisor
Fe
b
ru
ar
y
8,
20
08
ii
20
08
Contents
1 Goals for Physics 237
1
8,
2 Instructions for Physics 237
2.1 Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . .
Refraction of
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4 Exercise on Exploration of
4.1 Purpose . . . . . . . . .
4.2 Introduction . . . . . . .
4.3 Procedure . . . . . . . .
4.3.1 Preparation . . .
4.3.2 Investigation . .
4.4 Recap . . . . . . . . . .
Wave
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Optics Effects
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3 Exercise on Reflection and
3.1 Purpose . . . . . . . . .
3.2 Introduction . . . . . . .
3.3 Theory . . . . . . . . . .
3.3.1 Reflection . . . .
3.3.2 Refraction . . . .
3.4 Procedure . . . . . . . .
3.4.1 Apparatus . . . .
3.4.2 Preparation . . .
3.4.3 Experimentation
3.4.4 Analysis . . . . .
3.5 Bonus . . . . . . . . . .
3.6 Recap . . . . . . . . . .
3.7 Summary . . . . . . . .
3.8 Template . . . . . . . . .
Light
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iv
CONTENTS
4.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
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29
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7 Exercise on The Brewster Angle
7.1 Purpose . . . . . . . . . . . . . . . .
7.2 Theory . . . . . . . . . . . . . . . . .
7.2.1 Production of Polarized Light
7.3 Procedure . . . . . . . . . . . . . . .
7.3.1 Experimentation . . . . . . .
7.3.2 Analysis . . . . . . . . . . . .
7.4 Recap . . . . . . . . . . . . . . . . .
7.5 Summary . . . . . . . . . . . . . . .
7.6 Template . . . . . . . . . . . . . . . .
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35
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6 Exercise on The Polarimeter
6.1 Purpose . . . . . . . . . . .
6.2 Theory . . . . . . . . . . . .
6.3 Procedure . . . . . . . . . .
6.3.1 Apparatus . . . . . .
6.3.2 Experimentation . .
6.3.3 Analysis . . . . . . .
6.4 Recap . . . . . . . . . . . .
6.5 Summary . . . . . . . . . .
6.6 Template . . . . . . . . . . .
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8,
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20
08
5 Exercise on Malus’ Law
5.1 Purpose . . . . . . . . .
5.2 Theory . . . . . . . . . .
5.3 Procedure . . . . . . . .
5.3.1 Apparatus . . . .
5.3.2 Experimentation
5.3.3 Analysis . . . . .
5.4 Recap . . . . . . . . . .
5.5 Summary . . . . . . . .
5.6 Template . . . . . . . . .
8 Exercise on Dispersion of Glass
45
8.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
8.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
CONTENTS
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10 Exercise on Newton’s Rings
10.1 Purpose . . . . . . . . . .
10.2 Theory . . . . . . . . . . .
10.2.1 Newton’s Rings . .
10.3 Procedure . . . . . . . . .
10.3.1 Preparation . . . .
10.3.2 Apparatus . . . . .
10.3.3 Experimentation .
10.3.4 Analysis . . . . . .
10.4 Recap . . . . . . . . . . .
10.5 Summary . . . . . . . . .
10.6 Template . . . . . . . . . .
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Interferometer
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9 Exercise on the Michelson
9.1 Purpose . . . . . . . . .
9.2 Theory . . . . . . . . . .
9.3 Procedure . . . . . . . .
9.3.1 Apparatus . . . .
9.3.2 Preparation . . .
9.3.3 Experimentation
9.3.4 Analysis . . . . .
9.4 Recap . . . . . . . . . .
9.5 Summary . . . . . . . .
9.6 Template . . . . . . . . .
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47
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8.4
8.5
8.6
Procedure . . . . . . . .
8.3.1 Preparation . . .
8.3.2 Experimentation
8.3.3 Analysis . . . . .
Recap . . . . . . . . . .
Summary . . . . . . . .
Template . . . . . . . . .
8,
8.3
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11 Exercise on Generalized Least Squares Fitting
11.1 Purpose . . . . . . . . . . . . . . . . . . . . . .
11.2 Introduction . . . . . . . . . . . . . . . . . . . .
11.3 Theory . . . . . . . . . . . . . . . . . . . . . . .
11.3.1 Non-linear . . . . . . . . . . . . . . . . .
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vi
CONTENTS
72
73
75
83
83
85
85
86
12 Exercise on Sketching Functions and Extracting Parameters
12.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3.1 Critical Points . . . . . . . . . . . . . . . . . . . . . . .
12.3.2 Limits and Asymptotes . . . . . . . . . . . . . . . . . .
12.3.3 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3.4 Modified Functions . . . . . . . . . . . . . . . . . . . .
12.3.5 Substitution . . . . . . . . . . . . . . . . . . . . . . . .
12.4 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . .
12.4.2 Investigation . . . . . . . . . . . . . . . . . . . . . . .
12.4.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
12.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.6 Template . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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11.6
11.7
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11.4
11.5
11.3.2 Linear . . . . .
11.3.3 Goodness of fit
Example . . . . . . . .
Procedure . . . . . . .
11.5.1 Investigation .
11.5.2 Analysis . . . .
Summary . . . . . . .
Template . . . . . . . .
Fe
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13 Exercise on the Solver Spreadsheet Feature
13.1 Purpose . . . . . . . . . . . . . . . . . . . . .
13.2 Introduction . . . . . . . . . . . . . . . . . . .
13.3 Theory . . . . . . . . . . . . . . . . . . . . . .
13.3.1 Uncertainties in Parameters . . . . . .
13.3.2 Solver Example: Least Squares Fitting
13.3.3 Getting the parameter uncertainties . .
13.3.4 Solver in Excel TM . . . . . . . . . . .
13.4 Procedure . . . . . . . . . . . . . . . . . . . .
13.4.1 Preparation . . . . . . . . . . . . . . .
13.4.2 Investigation . . . . . . . . . . . . . .
13.4.3 Analysis . . . . . . . . . . . . . . . . .
13.5 Recap . . . . . . . . . . . . . . . . . . . . . .
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106
CONTENTS
vii
13.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
13.7 Template . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
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109
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A Aligning the Spectrometer
A.1 Introduction . . . . . . . . . . . .
A.1.1 Levelling the Spectrometer
A.1.2 Telescope Adjustment . .
A.1.3 Collimator Alignment . . .
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14 Exercise on the Goal Seek Spreadsheet Feature
14.1 Purpose . . . . . . . . . . . . . . . . . . . . . . .
14.2 Introduction . . . . . . . . . . . . . . . . . . . . .
14.3 Theory . . . . . . . . . . . . . . . . . . . . . . . .
14.3.1 Uncertainties in Parameters . . . . . . . .
14.3.2 Example: Non-Invertible Functions . . . .
14.3.3 Goal Seek in Excel TM . . . . . . . . . . .
14.4 Procedure . . . . . . . . . . . . . . . . . . . . . .
14.4.1 Preparation . . . . . . . . . . . . . . . . .
14.4.2 Investigation . . . . . . . . . . . . . . . .
14.4.3 Analysis . . . . . . . . . . . . . . . . . . .
14.5 Recap . . . . . . . . . . . . . . . . . . . . . . . .
14.6 Summary . . . . . . . . . . . . . . . . . . . . . .
14.7 Template . . . . . . . . . . . . . . . . . . . . . . .
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121
121
121
122
123
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viii
CONTENTS
20
08
List of Figures
Reflection of Light at a Mirror . . . . . . . . . . . . . . . . . . 9
Virtual Image Produced by a Mirror . . . . . . . . . . . . . . 9
Refraction of Light as it Passes from one Medium to Another . 10
Refraction of Light Passing Through a Glass Slab . . . . . . . 12
7.1
7.2
7.3
7.4
The Polarization of Light . . . . . . . . . . . . .
Brewster’s Law . . . . . . . . . . . . . . . . . .
Polarization by Successive Reflection . . . . . .
Polarization by Reflection and by Transmission
8.1
8.2
8.3
Path of Light Through a Prism in Air . . . . . . . . . . . . . . 46
Orientation of Prism to Find Prism Angle . . . . . . . . . . . 48
Orientation to Find Angle of Minimum Deviation . . . . . . . 50
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3.1
3.2
3.3
3.4
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36
37
39
40
10.1 Plano–Convex Lens . . . . . . . . . . . . . . . . . . . . . . . . 65
10.2 Newton’s Rings Setup . . . . . . . . . . . . . . . . . . . . . . 66
Plot of Sample Data . . . . . . . . . . .
Graph of Fit to A + B/x . . . . . . . . .
Graph of Fit to A + B/x + C/x2 . . . .
Graph of Fit to A + B/x + C/x2 + D/x3
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11.1
11.2
11.3
11.4
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75
76
80
83
12.1 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
12.2 Malus’ law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
13.1 Solver parameters . . . . . . . . . . . . . . . . . . . . . . . . . 103
13.2 Solver options . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
13.3 Solver results . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
14.1 Searching Near Root . . . . . . . . . . . . . . . . . . . . . . . 111
x
LIST OF FIGURES
14.2 Searching Near Local Minimum . . . . . . . . . . . . . . . . . 113
14.3 Goal Seek parameters . . . . . . . . . . . . . . . . . . . . . . . 116
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A.1 Telescope Adjustment . . . . . . . . . . . . . . . . . . . . . . 122
A.2 Collimator Adjustment . . . . . . . . . . . . . . . . . . . . . . 123
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Light reflecting from a mirror .
Light refracting through a slab .
Calculated index of refraction .
Light refracting through a prism
Calculated refraction angles . .
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4.1
4.2
4.3
Spectrum data . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Illumination data . . . . . . . . . . . . . . . . . . . . . . . . . 22
Grating data . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.1
Polarization effect . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.1
6.2
Specific Rotations for Various Sugars . . . . . . . . . . . . . . 31
Type of sugar: . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7.1
7.2
Sketch of spectrometer scales . . . . . . . . . . . . . . . . . . 44
Angle where maximum polarization is observed . . . . . . . . 44
8.1
8.2
8.3
Calculation of Prism Angle . . . . . . . . . . . . . . . . . . . . 53
n vs Angle of minimum deviation . . . . . . . . . . . . . . . . 54
n vs Wavelength . . . . . . . . . . . . . . . . . . . . . . . . . 55
9.1
Mercury lamp data . . . . . . . . . . . . . . . . . . . . . . . . 61
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17
17
18
18
18
3.1
3.2
3.3
3.4
3.5
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List of Tables
10.1 ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
11.1
11.2
11.3
11.4
Sample Data . . . . . . . . . . .
Data for Fit to A + B/x . . . .
LINEST 2 parameter output . .
Data for Fit to A + B/x + C/x2
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75
76
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78
xii
LIST OF TABLES
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78
81
81
82
84
86
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11.5 LINEST 3 parameter output . . . . . . .
11.6 Data for Fit to A + B/x + C/x2 + D/x3
11.7 LINEST 4 parameter output . . . . . . .
11.8 Comparisson of Fit Parameters . . . . .
11.9 F–Distribution Table (α = 0.05) . . . . .
11.10Fit comparison . . . . . . . . . . . . . .
12.1 Extracting parameters . . . . . . . . . . . . . . . . . . . . . . 95
13.1
13.2
13.3
13.4
Least Squares Fit data . . . . .
Finding parameter sensitivities .
Fit values . . . . . . . . . . . .
Parameter uncertainties . . . .
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100
101
107
107
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14.1 Helium wavelengths from curve . . . . . . . . . . . . . . . . . 119
Goals for Physics 237
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Chapter 1
• polarization
• dispersion
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• interference
8,
Thes labs will familiarize you with some basic optics concepts such as:
It may involve other areas such as:
• data analysis
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• technical communication
Goals for Physics 237
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2
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Chapter 2
2.1
Expectations
8,
Instructions for Physics 237
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By this stage of your university career, there are certain things expected of
you. Some of them are as follows:
• You are expected to come to the lab prepared. This means first of all
that you will attend all lectures so that you have all of the information
you need to do the labs. After you have been told what lab you will
be doing, you should read it ahead and be clear on what it requires.
You should bring the lab manual, lecture notes, etc. with you to every
lab. (Of course you will be on time so you do not miss important
information and instructions.)
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• You are expected to be adaptable and use common sense. In labs it
is often necessary to change certain details (eg. component values or
procedures) at lab time from what is written in the manual. You should
be alert to changes, and think rationally about those changes a react
accordingly.
• You are expected to value the time of instructors and lab demonstrators.
This means that you make use of the lab time when it is scheduled,
and try to make it as productive as possible. This means not arriving
late or leaving early and then seeking help at other times for what you
missed.
4
Instructions for Physics 237
• You are expected to use all of the resources at your disposal. This includes everything in the lab manual, textbooks for other related courses,
the library, etc.
20
08
• You are expected to be professional about your work. This means
meeting deadlines, understanding and meeting requirements for labs,
reports, etc. This means doing what should be done, rather than what
you think you can get away with. This means proofreading reports for
spelling, grammar, etc. before handing them in.
8,
• You are expected to collect your own data. This means that you perform experiments with your partner and no one else. If, due to an
emergency, you are forced to use someone else’s data, you must explain
why you did so and explain whose data you used. Otherwise, you are
committing plagiarism.
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• You are expected to do your own work. This means that you prepare
your reports with your partner and no one else. If you ask someone
else for advice about something in the lab, make sure that anything you
write down is based on your own understanding. If you are basically
regurgitating someone else’s ideas, even in your own words, you are
committing plagiarism. (See the next point.)
• You are expected to understand your own report. Even if you divide
work, such as writing up labs, with your partner, you are each still
responsible for all of the information in your report.
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• You are expected to be organized This includes recording raw data with
sufficient information so that you can understand it, keeping proper
backups of data, reports, etc., hanging on to previous reports, and so
on. It also means starting work early so there is enough time to clarify
points, write up your report and hand it in on time.
• You are expected to act on feedback from instructors, markers, etc. If
you get something wrong, find out how to do it right and do so.
• You are expected to actively participate in your own education. This
means that in the lab, you do not leave tasks to your partner because
you do not understand them. This means that you try and learn how
2.1 Expectations
5
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and why to do something, rather than merely finding out the result of
doing something.
Instructions for Physics 237
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6
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Chapter 3
3.1
Purpose
8,
Exercise on Reflection and
Refraction of Light
3.2
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This exercise is designed for the demonstration of the law of reflection of light
from a mirror surface and the location of the image of the object formed by
a mirror. Further, the law of refraction of light is demonstrated and the path
of light rays through a glass prism is studied. The index of refraction for the
glass used is measured as well.
Introduction
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This exercise will develop skills in dealing with uncertainties in trigonometric
functions. The focus is on geometric optics.
3.3
3.3.1
Theory
Reflection
When a beam of light in air strikes a medium with different electrical properties (here the metallic surface of a mirror, or glass) some of the light is
reflected at the interface. The fraction of light reflected depends on the electrical properties of the material. This paper reflects about 60% of the light
8
Exercise on Reflection and Refraction of Light
incident upon it, while a polished mirror surface reflects more than 90% of
incident light.
20
08
If the material surface is relatively rough on the scale of the wavelength of
visible light, such as this paper, the light is reflected in all directions: a beam
of light is diffused by the surface upon reflection. This is the reason that you
don’t see a clear reflected image of the lights on the ceiling from this paper.
On the other hand, for very smooth surfaces such as that of a mirror the
reflected rays are not diffused at all, so that the eye can focus them and form
a real image of the reflected object on its retina. Such reflection is termed
regular or specular.
8,
The law for reflection from specular surfaces has been known for centuries;
it states that
the angle of incidence (i) equals the angle of reflection (r)
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as they are defined in Figure 3.1. ( i and r are measured relative to the
normal to the reflecting surface at the point where reflection occurs.)
When an optical element such as a mirror is placed in the line of sight, rays
of light entering the eye from the mirror appear to have come from a point
behind the mirror. We say that a virtual image of the object exists behind
the mirror. The word virtual means the image is not real, that the light rays
don’t actually come from it, but they appear to do so. On the other hand,
rays from the object are reflected from the mirror and enter the eye where
they are focused on the retina creating a real image of the object there.
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As in Figure 3.2, two rays from a “point” object are reflected from a mirror
to the eye. If the path of the actual reflected rays is traced behind the mirror
(as dashed lines) they appear to have come from an object behind the mirror
which is the virtual image of the object.
3.3.2
Refraction
It was stated previously that when a ray of light in air strikes a medium with
different electrical properties some of the light is reflected at the interface;
the light that isn’t reflected is refracted into the next material (glass in our
case).
3.3 Theory
9
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surface normal
incident ray
reflected ray
r
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mirror surface
Figure 3.1: Reflection of Light at a Mirror
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normals
eye
object
virtual image
mirror surface
virtual rays
Figure 3.2: Virtual Image Produced by a Mirror
10
Exercise on Reflection and Refraction of Light
i
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air
normal
incident ray
na
reflected ray
ng
refracted ray
8,
glass
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r
Figure 3.3: Refraction of Light as it Passes from one Medium to Another
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When light passes through the interface between two media into the second,
its direction is changed in accordance with Snell’s Law of Refraction which
relates the angles of incidence and refraction and the indices of refraction of
each of the media.
Consider the interface between air and glass in Figure 3.3 which defines the
angles of incidence (i) and refraction (r) as measured with respect to the
normal to the surface at the point of incidence.
The index of refraction or refractive index (n) for a material is the ratio
of the speed of light in a vacuum (v0 ) to the speed of light in the material
(vm ) whence n = v0 /vm .
Snell’s Law states that
na sin i = ng sin r
3.4 Procedure
11
where na is the refractive index of air, and to a very good approximation
na ≈ 1
sin i = ng sin r
where ng is the refractive index of the glass.
20
08
We can thus rewrite Snell’s Law for air to glass as
For the case where the light passes through a sheet of glass with parallel sides,
consider Figure 3.4. The ray is incident in air at angle i and is refracted to
angle r at the first interface;
(sin i = ng sin r)
8,
At the second interface the ray is incident at angle i0 and is refracted to air
at angle r0 ;
(ng sin i0 = sin r0 )
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Note: The uncertainty in sin θ is given by
∆ sin θ ≈ |cos θ| ∆θ
for θ in radians.
Note: The uncertainty in Arcsin(x) is given by
∆Arcsin(x) ≈ √
1
∆x
1 − x2
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(in radians). Note that this gets very big as x approaches 1.
3.4
Procedure
3.4.1
Apparatus
• Plane mirror (front-surfaced)
• Thick glass prism
• Pin-board and hat pins
• Protractor, straight-edge,plane sheets of clean 81/2” × 11” paper.
12
Exercise on Reflection and Refraction of Light
normal
incident ray
glass
ng
i0
r
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08
i
air
emergent ray
air
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r0
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Figure 3.4: Refraction of Light Passing Through a Glass Slab
3.4.2
Preparation
Since this is the first week of labs, there are no pre-lab requirements.
3.4.3
Experimentation
In-lab Tasks
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The in-lab tasks are included with each part.
Part 1: Angles of Incidence and Reflection
Please do not touch the mirror surfaces.
Note that the mirror is “front-surfaced”; light is reflected from its front surface In household mirrors, the silvering is placed on the back of a sheet of
glass and sealed in so that the reflecting surface will last for years and years.
In front-surfaced mirrors, the light rays don’t have to pass through the glass
first in order to be reflected.
3.4 Procedure
13
20
08
1. Draw a line down the centre of a piece of clear paper and place it on
the pin-board. Set the mirror surface against this line near the centre
of the page, standing on edge. On the front side of the mirror, off to
one side, stick a hat-pin vertically into the pin-board to serve as an
object. Viewing the image of this pin in the mirror, place two hat pins
along the direction of a reflected ray. Keep the two pins as far apart as
possible.
2. Remove the pins, and circle these three holes in the paper.
3. Repeat this procedure once more for another viewing direction, using
something other than circles to mark the other set of holes.
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4. From each of the two sets of pin-holes you have the direction of the
reflected ray, which you can extend to find the point of reflection at the
mirror. From the point of reflection and the object pin hole, you can
draw the incident ray direction. Remove pins, mirror and sheet, and
draw in all incident and reflected rays and the surface normal at each
point of reflection. Measure the incident and reflected angles for each
of the rays and enter them in Table 3.1.
IT1: Show your diagram to the lab instructor.
Part 2: Virtual Images
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1. Begin with paper, line, and mirror as in the previous part. Draw a
large letter V somewhere in front of the mirror. Mark the two ends of
the V in some way (such as A, B, C) so that you can tell them apart
in the reflection. Hint: Make the two arms of the V different lengths
and it will make them easier to distinguish.
2. Place a pin in one of the strategic points of the (object) letter, (say
A), and find the incident and reflected rays in two directions, labelling
them (such as A1 for the points in one direction, and and A2 for the
two points in the other direction).
3. Repeat with hat-pins placed in the other strategic points of the object
letter.
14
Exercise on Reflection and Refraction of Light
20
08
4. Remove pins and mirror, and on the sheet extend each pair of reflected
rays back behind the mirror as dashed lines, to where they intersect.
The three intersection points thus gained mark the strategic points of
the virtual image of the letter used as an object. Draw in this virtual
image using dotted lines.
IT2: Show your diagram to the lab instructor.
Part 3: Refraction through Parallel Surfaces
8,
1. Draw a line down the centre of a piece of paper as before. With paper
on the pin board, place the glass prism standing on one of its long
edges, against the line, centred on the page.
2. Draw a line along the back edge of the parallel-sided glass block on the
page.
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3. Place a pin in the paper behind the prism off to one side. Observe from
a position which will give a large angle of incidence. Use another pin
on the same side of the paper so that you can map the incident ray.
4. With 2 more pins, map out the emergent ray. Remove the pins and
label the 4 points.
5. Repeat the previous two steps for another object point.
6. Remove the prism.
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7. Draw the incident and emergent rays for both cases, and then draw in
the rays refracted through the glass as well. Measure the angles i, r,
i0 and r0 for both rays and record them in Table 3.2. Remember that
you’ll have to draw in normal lines to measure the angles, or else use
(90 − angle) if you measure from the glass surface. You may have to
extend the rays inside the glass block in order to measure them.
Note that the emergent ray should be exactly parallel to the incident
ray for each case.
IT3: Show your diagram and Table 3.2 to the lab instructor.
3.4 Procedure
15
Part 4: Refraction Through a Prism
20
08
1. Place the prism flat on one of its broad faces on a sheet of paper and
draw in its outline carefully.
2. Place 2 pins along a path to the prism with an incident angle i about
45◦ .
3. Look through the side of the prism where the refracted ray emerges,
and place 2 pins along the line of the emergent ray.
8,
4. Remove the prism. Draw lines connecting both sets of pins with the
faces of the prism, and then draw the line though the prism which joins
them and thus trace the ray path through the prism experimentally.
5. Measure the incident, refracted, and emergent angles and put them in
Table 3.4.
3.4.4
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IT4: Show your diagram and Table 3.4 to the lab instructor.
Analysis
Post-lab Questions
Q1: Calculate ng and ∆ng from the data in Part 3, placing the results in
Table 3.3.
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1. Measure the prism angle from your diagram of Part 4, and use it along
with the average n from Part 3 to calculate each of the angles in Table 3.5. Show your calculations.
2. Draw in the ray based on your calculations as it passes through the
prism to the other face. Higlight this line to distinguish it from the
experimental one.
3. Draw in the ray as it leaves the second surface, based on your calculations. Higlight this line to distinguish it from the experimental one.
4. Put all the calculations on the sheet with the prism outline and the
calculated ray path.
16
Exercise on Reflection and Refraction of Light
Post-lab Tasks
T1: Photocopy your diagram and your calculations and hand them in.
Bonus
20
08
3.5
Include uncertainties in the last part. Note that in each step, the uncertainty
in an angle makes an uncertainty in the position of the next refraction.
3.6
Recap
• Draw ray diagrams for
8,
By the end of this exercise, you should know how to :
– reflection at a plane surface
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– refraction at a plane interface
3.7
Summary
Item
Pre-lab Questions
In-lab Questions
Post-lab Questions
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Pre-lab Tasks
In-lab Tasks
Post-lab Tasks
Number
0
0
1
0
4
1
Received
weight (%)
0
0
20
0
60
20
3.8 Template
Template
My name:
My student number:
My partner’s name:
My other partner’s name:
Today’s date:
ray
Angle (degrees)
i
r
1
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08
3.8
17
8,
2
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Table 3.1: Light reflecting from a mirror
ray
i
Angle (degrees)
r
i0
r0
1
2
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Table 3.2: Light refracting through a slab
18
Exercise on Reflection and Refraction of Light
ray
From incident
n
∆n
From emergent
n
∆n
20
08
1
2
average
(n̄)
uncertainty
(∆n̄)
8,
Table 3.3: Calculated index of refraction
Angle (degrees)
r
i0
r0
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ray
i
1
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Table 3.4: Light refracting through a prism
ray
r
∆r
Angle (degrees)
i0
∆i0
r0
1
Table 3.5: Calculated refraction angles
∆r0
20
08
Chapter 4
4.1
Purpose
8,
Exercise on Exploration of
Wave Optics Effects
4.2
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The purpose of this exercise is to introduce you to optical effects all around
us.
Introduction
The focus is on wave optics.
Procedure
4.3.1
Preparation
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4.3
Pre-lab Questions
PQ1: What is the range of wavelengths which are visible? (Give reference.)
PQ2: What is the reccomended illumination (in lux) for work areas? (Give
reference.)
20
Exercise on Exploration of Wave Optics Effects
4.3.2
Investigation
In-lab Tasks
The in-lab tasks are included with each part.
20
08
Part 1: Spectrum analysis
1. Observe and sketch the spectrum for daylight and two different types
of light sources seen in the building. (Also note if either source is
polarized, and if so, in what direction.)
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Part 2: Illumination
8,
IT1: Put the results in Table 4.1 and explain your findings to the lab instructor.
1. Measure the illumination in three different work areas in the science
building.
IT2: Put the results in Table 4.2 and explain your findings to the lab instructor.
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Part 3: Diffraction
1. Using a laser, observe the diffraction patterns for 3 gratings, and see
how the dot spacing depends on the grating spacing.
IT3: Put the results in Table 4.3 and explain your findings to the lab instructor.
4.4 Recap
21
IT4: Demonstrate your findings to the lab instructor.
4.4
Recap
Summary
Item
Pre-lab Questions
In-lab Questions
Post-lab Questions
source
Received
0
4
0
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Pre-lab Tasks
In-lab Tasks
Post-lab Tasks
Number
2
0
0
4000
(Å)
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weight (%)
20
0
0
8,
4.5
20
08
By the end of this exercise, you should be familiar with some optical phenomena.
Table 4.1: Spectrum data
0
80
0
7000
polarized
(y/s/n)
22
Exercise on Exploration of Wave Optics Effects
illumination
(lux)
20
08
location
1
2
3
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8,
Table 4.2: Illumination data
grating spacing
(lines/inch)
dot spacing
(cm)
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1
2
3
Table 4.3: Grating data
20
08
Chapter 5
Exercise on Malus’ Law
Purpose
8,
5.1
The purpose of this exercise is to study Malus’ Law.
Theory
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5.2
When making quantitative measurements of polarization, it is useful to define
the degree of polarization, P
P =
Ik − I⊥
Ik + I⊥
(5.1)
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where Ik is the transmitted intensity with the analyzer axis parallel to some
fixed reference direction, and I⊥ is the transmitted intensity with the analyzer axis perpendicular to the reference direction. For example, if you are
measuring the degree of polarization produced by reflection from a surface,
the reference direction is the surface itself. Thus Ik is the intensity with the
analyzer axis parallel to the surface, and I⊥ is the intensity with the analyzer axis rotated through 90◦ . Clearly, the degree of polarization is zero for
unpolarized light, and ±1 for completely polarized light.
Since the polaroid filters are not 100% efficient (i.e., if unpolarized light is
used, the transmitted beam is not completely polarized), a correction factor,
known as the analyzer efficiency, α, must be introduced into Equation 5.1
if absolute polarization measurements are required. If the incident light has
intensity components (with respect to the reference direction) Ik and I⊥ ,
24
Exercise on Malus’ Law
and the analyzer is oriented so that its transmission axis is parallel to the
reference direction, the measured intensity is
Ikm = tk Ik + t⊥ I⊥
(5.2)
20
08
Now, rotating the analyzer through 90◦ , the transmission axis is perpendicular to the reference direction, and the measured intensity is
I⊥m = tk I⊥ + t⊥ Ik
(5.3)
where tk is the transmission coefficient of the analyzer along its transmission
axis, and t⊥ is the transmission coefficient perpendicular to the transmission
axis. Using Cramer’s rule, the true intensities are
tk Ikm − t⊥ I⊥m
t2k − t2⊥
8,
Ik =
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I⊥ =
tk I⊥m − t⊥ Ikm
t2k − t2⊥
(5.4)
(5.5)
Substituting into Equation 5.1.
m
m
tk + t⊥ Ik − I⊥
P =
tk − t⊥ Ikm + I⊥m
Pm
=
α
(5.6)
(5.7)
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where we have defined
α=
Pm =
tk − t⊥
tk + t⊥
Ikm − I⊥m
Ikm + I⊥m
(5.8)
(5.9)
Thus, the measured polarization Pm is always less than or equal to the true
polarization, since α ≤ 1.
The analyzer efficiency can be measured in one of two ways — determining the measured polarization of completely polarized light, as produced by
5.3 Procedure
25
reflection from a surface at the Brewster angle, or, by using two identical
polaroid filters and unpolarized light. In the first case, the incident light has
a true polarization of 1. Then the analyzer efficiency is just the measured
polarization,
α = Pm
(5.10)
20
08
In the second case, the first filter is used to produce partially polarized light,
and the second is used to analyze it. Since the light is initially unpolarized,
it can be shown that the analyzer efficiency is
p
(5.11)
α = Pm
Procedure
5.3.1
Apparatus
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5.3
8,
where the square root results from the fact that the filters are identical, so
the transmission through both is equal to the product the transmissions of
both.
The following pieces of equipment are required for this experiment:
• a pair of polaroid filters,
• light source,
• photocell, milli-microammeter,
5.3.2
Experimentation
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Determination of Analyzer Efficiency
The photocell and the milli-microammeter will be used to measure the light
intensity in this experiment. In each case, the ambient (background) intensity
will have to be measured. Best results will be obtained if you work in a
darkened room. This background intensity will have to be subtracted from
your measured intensities before calculating the degree of polarization.
1. Mount the white light source and the photocell on an optical bench, and
connect the ammeter to the photocell. Place the polarizer–analyzer pair
in front of the photocell, as close to it as possible, without interfering
with the rotation of the analyzer.
26
Exercise on Malus’ Law
2. Block off the light source and remove the polarizer. Check the background intensity every 10◦ . Is the background light measureable? (If it
is, then you’ll need to record it.) If it is measureable, is the background
light unpolarized?
20
08
3. Now unblock the light source and again measure the light intensity as a
function of the analyzer angle every 10◦ . Plot a graph of true intensity
as a function of angle. Does your light source produce unpolarized
light?
4. Place the polarizer in front of the analyzer and align the transmission
axes. Rotate the analyzer in 10◦ steps, and measure the light intensity
and background intensity.
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8,
5. Locate the positions of the maximum and mininum intensities, and
record the angles and photocurrents. Using the maximum (Ik ) and
minimum (I⊥ ), calculate the measured polarization, and the analyzer
efficiency, assuming unpolarized incident radiation.
Inlab Questions
IQ1: What are the measured polarization and the analyzer efficiency? Show
your calculations.
Inlab Tasks
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IT1: Summarize your findings, including measured polarization and analyzer
efficiency, to the lab instructor.
5.3.3
Analysis
Determination of Analyzer Efficiency
1. Plot a graph of true intensity vs angle.
2. Plot a graph of true intensity vs the sine of the angle.
3. Plot a graph of true intensity vs the square of the sine of the angle.
5.4 Recap
27
Postlab Questions
Q1: Discuss whether either of the last two graphs appears linear.
20
08
Postlab Tasks
T1: Hand in your graphs and the question answers.
5.4
Recap
5.5
Summary
Number
0
1
1
Received
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Item
Pre-lab Questions
In-lab Questions
Post-lab Questions
8,
By the end of this exercise, you should know how to measure polarization
and determine polarizer efficiency.
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Pre-lab Tasks
In-lab Tasks
Post-lab Tasks
0
1
1
weight (%)
0
30
20
0
30
20
28
Template
My name:
My student number:
My partner’s name:
My other partner’s name:
Today’s date:
intensity
angle
intensity
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8,
angle
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08
5.6
Exercise on Malus’ Law
Table 5.1: Polarization effect
20
08
Chapter 6
Exercise on The Polarimeter
Purpose
8,
6.1
6.2
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The purpose of this lab is to investigate optically active materials and the
polarimeter.
Theory
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Some optically active materials can rotate the plane of polarization when
polarized light is passed through them. The amount of rotation is proportional to the optical path length the light must traverse. Optical materials
that rotate the plane (looking in the direction of propagation) to the right
are called dextro-rotary, and those that rotate the plane to the left are called
leuvo-rotary. This has to do with the fact that the molecules of the material
have a handedness, or asymmetry, in their molecular structure (you would
be able to distinguish between the molecule and a mirror image of it).
6.3
Procedure
6.3.1
Apparatus
The following pieces of equipment are required for this experiment:
• polarimeter,
• Na arc lamp,
30
Exercise on The Polarimeter
• various sugar solutions.
6.3.2
Experimentation
20
08
The Polarimeter
1. Choose a sugar to use, and record the name in Table 6.2.
2. With just plain water in the tube, rotate the polarizer of the instrument
to determine the angle of the plane of polarization, using the Na lamp
as the light source. Record this in Table 6.2.
8,
3. In 100 mls of water, dissolve 1 gm of sugar. Put this solution into
the tube of the polarimeter. Rotate the polarizer of the instrument to
determine the amount of rotation of the plane of polarization.
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4. If the change in the angle of the plane of polarization due to the 1
gm of sugar was significant, record this in Table 6.2 and then repeat
the measurement for 2, 3 and 4 gms of sugar. Otherwise, repeat this
measurement using bigger increments in the amount of sugar.
Inlab Questions
IQ1: Is the sugar you are examining optically active? If so, which direction
does it rotate the plane of polarization of light passing through? (Remember
the light goes from the source to your eye to determine the direction of
rotation.) Is the change linear with concentration?
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Inlab Tasks
IT1: Summarize your results to the lab supervisor.
6.3.3
Analysis
The Polarimeter
1. Plot the rotation produced as a function of concentration and calculate
the specific rotation, α, in degrees/cm of path per gram of sugar/100
mls water and compare this to the value given in the table below.
6.3 Procedure
31
Sugar
α
/cm
+6.6412
+5.25
-8.85
+13.848
-5.14
-9.2
sucrose
dextrose
levulose
maltose
glucose
fructose
20
08
◦
Table 6.1: Specific Rotations for Various Sugars
• All values are at 20◦ C.
Specific rotation:1 “If there are
8,
• All values are for the sodium D line.
• n grams of active substance in
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• v cubic centimetres of solution
• and the light passes through ` centimetres,
• r being the observed rotation in degrees,
the specific rotation
α=
rv
n`
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is given”.
Postlab Questions
Q1: Does your calculated specific rotation agree with the value from the
table? Show your calculations and discuss.
Postlab Tasks
T1: Hand in a copy of your data, along with your graph and the question
answers.
1
The CRC Handbook 49, F-94 (1968)
32
Exercise on The Polarimeter
6.4
Recap
By the end of this exercise, you should know how to :
• read a meter with a Vernier scale
Summary
Item
Pre-lab Questions
In-lab Questions
Post-lab Questions
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Pre-lab Tasks
In-lab Tasks
Post-lab Tasks
Number
0
1
1
Received
8,
6.5
20
08
• determine specific rotation for a solution
0
1
1
weight (%)
0
30
20
0
30
20
6.6 Template
6.6
33
Template
Angle (degrees)
(◦ )
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8,
concentration
(g/100 ml)
20
08
My name:
My student number:
My partner’s name:
My other partner’s name:
Today’s date:
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Table 6.2: Type of sugar:
Exercise on The Polarimeter
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8,
20
08
34
20
08
Chapter 7
7.1
Purpose
8,
Exercise on The Brewster
Angle
7.2
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The purpose of this lab is to investigate polarization by reflection and the
Brewster angle.
Theory
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The ‘vibrations’ of a light wave are transverse, i.e., light is an electromagnetic wave whose electric and magnetic field vectors are perpendicular to the
direction of propagation. This can be seen from the solutions to Maxwell’s
equations for the electromagnetic field. The transverse nature of light waves
is illustrated diagrammatically Figure 7.1a , in which the radial arrows represent vibrations taking place in all directions in a plane perpendicular to
the line of sight. For the purposes of analysis, it is convenient to describe the
transverse vibrations in terms of components resolved along two mutually
perpendicular directions, XX and YY in Figure 7.1a .
In the case of ordinary, unpolarized light, energy is distributed uniformly
among transverse vibrations in all possible directions, and the perpendicular
components are equal, regardless of the chosen directions of resolution.
In the case of completely polarized light, only one plane of vibration exists.
This situation is illustrated in Figure 7.1b , where a polarizing device has
been used to select only the vertical components of all transverse vibrations
Exercise on The Brewster Angle
(b)
(c)
8,
(a)
20
08
36
Figure 7.1: The Polarization of Light
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in the incident light. It can be shown that the intensity of the polarized light
in Figure 7.1b is one-half of the original, unpolarized intensity.
Since the human eye is cannot distinguish between polarized and unpolarized light, we require a polarizing device; an instrument that will transmit
vibrations that take place only along one axis, known as the transmission
axis. If the observed intensity remains the same for all orientations of the
axis, the light is unpolarized. If (virtually) no light is transmitted for a
particular orientation of the polarizer, the incident light is completely plane
polarized, as shown in Figure 7.1c . Because the second polarizer in Figure 7.1c is used to analyze the polarization of light incident upon it, it is
referred to as an analyzer. The plane of polarization of the light can then
readily be determined by noting that the transmission axis of the analyzer is
perpendicular to it when no light is transmitted.
7.2.1
Production of Polarized Light
Polarizing Screens
The simplest method of producing polarized light makes use of a polarizing screen, composed of either natural or artificially prepared crystals that
37
8,
20
08
7.2 Theory
Figure 7.2: Brewster’s Law
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possess the property of being able to transmit only the components of the
transverse vibrations parallel to a specific direction in the crystal. Tourmaline is an example of a natural crystal which does this, but it is obtainable
only small and imperfect specimens. A commercial product marketed under
r is well suited to the construction of polarizing
the trade name of Polaroid screens of any size. It consists of a layer of iodo-quinine sulfate (herapathite)
crystals suitably oriented and imbedded in a thin sheet of transparent celluloid. While the polarization produced by these screens is practically complete
throughout most of the visible spectrum, the shortest violet and longest red
ends are only partially polarized. Since the eye (and most optical detectors designed to work in the visible) are not particularly sensitive to these
wavelengths, this limitation is not serious for most practical applications.
Polarization by Reflection
When a ray of light is incident on a smooth surface, the reflected component
is partially polarized, with the direction of maximum intensity parallel to
the surface (i.e., perpendicular to the plane of incidence). If the surface is
that of a transparent medium, the transmitted component is also partially
polarized, but with the direction of maximum intensity parallel to the plane
of incidence. This is illustrated in Figure 7.2 .
38
Exercise on The Brewster Angle
Of course, the reflected and transmitted rays are described by:
i = i0
and
(7.1)
20
08
sin i = n sin r
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8,
where i is the angle of incidence, i0 is the angle of reflection, r is the angle of
refraction, and n is the index of refraction of the medium.
It was discovered by Malus in 1808 that, in the case of non-metallic surfaces, when the angle of incidence is such that the angle between the reflected
and transmitted rays is 90◦ , the reflected ray is completely polarized parallel
to the surface. The particular angle of incidence for which this occurs is
called the polarizing angle, and it depends on the index of refraction n.
For glass, the polarizing angle is about 57◦ .
The required angle of incidence can be derived from the geometry of
Figure 7.2 . According to Malus’ Law, complete polarization of the reflected
ray occurs when i is such that the angle between the reflected ray and the
transmitted ray is 90◦ . Then it follows from the geometry that
i + r = 90◦
or
r = 90◦ − i
Using this in Equation 7.1
sin i = n sin r = n sin (90◦ − i) = n cos i
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or
n = tan i
and
i = arctan n
(7.2)
Equation 7.2 is known as Brewster’s Law, and the required angle of incidence is also known as the Brewster angle. At angles of incidence other
than Brewster’s angle, the reflected light is partially polarized.
As mentioned previously, the transmitted component is also partially polarized. By means of transmission through a number of parallel plates, the
repeated reflection progressively increases the degree of polarization of the
39
20
08
7.2 Theory
8,
Figure 7.3: Polarization by Successive Reflection
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transmitted ray, until, (after 10 or twelve reflections), the transmitted ray
is, for all practical purposes, completely polarized. From what has been
said, it should be clear that the plane of polarization of the transmitted ray
is perpendicular to that of the reflected ray. This process is illustrated in
Figure 7.3 .
Thus, a stack of glass plates inclined at the Brewster angle with respect
to the incident ray can be used as a polarizer in either of two ways, as shown
in Figure 7.4 .
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Polarizing screens are extensively used in photography for the elimination
of unwanted reflections and for lighting control. Anti–glare glasses are made
with Polaroid to reduce the glare produced by specular reflections from water,
hard-surface roads, etc.
How would you orient the transmission axis of the glasses to accomplish
this?
Polarization by reflection can be used to construct analyzers to operate
in spectral regions (such as the extreme ultraviolet) where the light would
be completely absorbed by any transmission through a medium. These are
usually made of a series of front surface mirrors, (gold or aluminum plated),
and produce polarizations of up to 99% , with about 10 to 20% of the incident
light intensity emerging.
Exercise on The Brewster Angle
8,
20
08
40
Figure 7.4: Polarization by Reflection and by Transmission
Procedure
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7.3
7.3.1
Experimentation
Apparatus
The following equipment is required for this experiment:
r filter
• Polaroid • spectrometer
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• glass slides
• holder for slides
• light source
Method
Measuring the Brewster Angle
1. Place the holder with the glass plates on the prism table of the spectrometer.
7.3 Procedure
41
2. Place a light source to send light through the collimator slit.
3. Hold a polarizing filter close to your eye and observe the reflected image
of the source, as shown in Figure 7.4a. Rotate the filter to find the
minimum transimission (i.e. the dimmest image of the slit).
20
08
4. Rotate the prism table, and repeat the previous instruction until you
find the angle at which the slit becomes invisible as the filter is rotated.
5. Move the telescope into position so that the slit image is lined up with
the crosshairs.
6. Measure the angle of the telecope from the spectrometer scale and
record it in Table 7.2.
8,
7. Rotate the prism table 10 or twenty degrees, and then repeat the last
three steps to determine the angle again. The difference between these
angles can be used to determine an uncertainty in the Brewster angle.
8. The Brewster angle should be 1/2 of the recorded angle.
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9. Look at the other side of the glass plates to see the image transmitted
through the plates.
10. Move the telescope into position so that the slit image is lined up with
the crosshairs, as before.
11. Measure the angle of the telecope from the spectrometer scale and
record it.
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Inlab Questions
IQ1: Why should the Brewster angle be half of the measured angle?
IQ2: Is the transmitted image of the slit at 90◦ to the reflected image?
Explain. (Hint: Look carefully at Figures 7.2 and 7.3.)
IQ3: If you turned the plates so that the reflected image was on the other
side of where it is now, how would your calculation of the Brewster angle
change?
IQ4: Write out one sample conversion from degrees, minutes, and seconds
(DMS) to decimal degrees (DD).
IQ5: Write out your calculation of the Brewster angle and its uncertainty.
42
Exercise on The Brewster Angle
Inlab Tasks
IT1: Sketch the experimental layout in your lab book.
20
08
IT2: Sketch the spectrometer scale for one measurement in Table 7.1 and
show the corresponding measurement to show that you can use the Vernier
scale correctly. (Be sure to show both the main and Vernier scales.)
IT3: Demonstrate the Brewster angle and summarize your findings to the
lab instructor.
Analysis
Postlab Questions
8,
7.3.2
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Q1: Calculate the index of refraction of the glass from Brewster’s Law. Show
your calculations.
Postlab Tasks
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T1: Hand in the question answers, along with a copy of your sketch of the
setup.
7.4
Recap
By the end of this exercise, you should know how to :
• find and measure the Brewster angle, and
• use it to calculate the index of refraction
To do this, you will have to learn how to read the spectrometer.
7.5 Summary
Summary
Item
Pre-lab Questions
In-lab Questions
Post-lab Questions
Received
0
3
1
weight (%)
0
40
10
0
40
10
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8,
Pre-lab Tasks
In-lab Tasks
Post-lab Tasks
Number
0
5
1
20
08
7.5
43
44
7.6
Exercise on The Brewster Angle
Template
8,
20
08
My name:
My student number:
My partner’s name:
My other partner’s name:
Today’s date:
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Table 7.1: Sketch of spectrometer scales
Trial
1
Angle (degrees)
2
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Table 7.2: Angle where maximum polarization is observed
20
08
Chapter 8
Exercise on Dispersion of Glass
Purpose
8,
8.1
The purpose of this exercise introduce you to the concept of dispersion.
Theory
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8.2
In this exercise we shall use the angle of minimum deviation to measure the
index of refraction of glass as a function of wavelength.
When a ray of light is obliquely incident on the interface separating two
transparent media of different indices of refraction, its direction changes. The
amount of deviation is described by Snell’s Law:
n1 sin θi = n2 sin θr
(8.1)
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where n1 is the index of refraction of medium 1, θi is the angle of incidence
(measured from the normal to the interface), n2 is the index of refraction of
medium 2, and θr is the angle of refraction.
For most transparent media, the angle of refraction increases with increasing frequency (for a fixed angle of incidence) — thus, blue light is refracted
through a much larger angle than red light. Since the definition of the index
of refraction is:
c
n=
(8.2)
v
where c is the speed of light in vacuum and v is the speed of light in a
medium, and since the index of refraction is always greater than or equal to
1 for any medium, it is clear that v is less than or equal to c.
46
Exercise on Dispersion of Glass
20
08
J
J
J
J
"
J
"
α
J
"
" T ""
J"
T"
J
-"
1
δ
JXXX
XXX min
J
X
XXX
J
z
X
J
J
n1
n
n1
J
J
Figure 8.1: Path of Light Through a Prism in Air
8,
The speed of light is related to frequency and wavelength by:
c = νλ
(8.3)
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and the frequency does not depend on the medium through which the light
travels. Thus it follows that the wavelength of light must change when it
enters a material medium.
Furthermore, the index of refraction of a material is not a constant, but
depends upon the frequency of the incident light. From measurements of the
index of refraction of a medium as a function of wavelength, we can construct
a dispersion curve by plotting a graph of n(λvac ) vs λvac , where λvac refers
to the vacuum wavelength of the incident light. (For the purposes of this
experiment, we shall assume that the wavelengths in air are identical to the
wavelengths in vacuum). The dispersion of the medium at a particular
wavelength λ0 is defined to be the slope of the dispersion curve at that
wavelength:
dn
(8.4)
D=
dλ λ0
Referring to Figure 8.1, whenever a ray of monochromatic light passes
through the prism, it is deviated through some angle δ. The value of this
angle depends on the angle of incidence at the first surface of the prism.
When the angle of incidence φi is equal to the angle of emergence φe , the
angle of deviation δ is a minimum, called the angle of minimum deviation
8.3 Procedure
47
δmin . The index of refraction of the glass, n, can then be found from:
n = n1
sin[(α + δmin )/2]
sin(α/2)
(8.5)
8.3
Procedure
8.3.1
Preparation
20
08
where α is the prism angle.
Pre-Lab Questions
8,
PQ1: Determine the equation for the uncertainty in n from Equation 8.5,
given α, δmin and the uncertainties ∆α and ∆δmin .
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PQ2: Determine the equation for the uncertainty in the average of x, i.e. x̄,
given the uncertainties in the individual measurements of x. (Hint: Check
the PC131 lab manual if you can’t remember.) (You’ll need this for later
calculations.)
PQ3: If n did not change with λ, i.e. if there were no dispersion, how would
the operation of a prism change?
8.3.2
Experimentation
Apparatus
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For this experiment you will need the following pieces of equipment:
• Spectrometer,
• 60◦ Prism,
• Sodium Lamp,
• Hydrogen, Helium and Mercury Geissler Tubes with Power Supply,
• Variable Intensity White–Light Source,
• assorted stands and clamps.
Exercise on Dispersion of Glass
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8,
α
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08
48
Figure 8.2: Orientation of Prism to Find Prism Angle
Method
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Aligning the Spectrometer
The spectrometer should be aligned already. If there are problems, see
Appendix A of the lab manual.
Measurement of Prism Angle
1. Place the prism on the prism table with the prism angle pointing toward
the collimator as in Figure 8.2.
8.3 Procedure
49
2. Place the light source in front of the collimator. Use a narrow collimator
slit in the following procedure. Note that the slit image can be seen
reflected in the prism faces.
20
08
3. Adjust the telescope so that the slit image is aligned with the cross–
hairs, read the vernier scale (30 part vernier every half–degree; thus
readings are in degrees and minutes), and record this reading (the
telescope angle).
Be as careful as possible when taking measurements, and be sure to record
values to the nearest minute of angle. Sloppy measurements will give you
meaningless results.
8,
4. Repeat this for the other prism face and enter your readings into Table 8.1.
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5. Repeat these measurements 3 times.
6. The prism angle is one half the difference between the two vernier
readings. From the three series of measurements recorded in Table 8.1,
determine the prism angle α for each measurement, and calculate the
average value of the prism angle. Include measurement uncertainties
in your final answer.
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Measuring the Angle of Minimum Deviation
If your values for δmin are not changing monotonically as expected, recheck
your measurements.
1. Place the sodium vapour lamp in front of the collimator slit.
2. Rotate the prism 90◦ so that now light will be refracted through it on
the way to the telescope. Observe the spectral lines and reduce the slit
width until they are sharp and narrow.
Exercise on Dispersion of Glass
20
08
50
Light source
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Prism Table
8,
Collimator
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Telescope
Observer
Figure 8.3: Orientation to Find Angle of Minimum Deviation
8.3 Procedure
51
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3. Rotate the prism table as shown in Figure 8.3 until the angle of minimum deviation is achieved for one of the Sodium–D lines (2 lines separated by 6 Å; these may be too close together to be clearly resolved —
if so, set the cross–hairs to the center of the doublet and use the mean
wavelength of 5893Å). Record the vernier reading in Table 8.2.
4. Repeat this measurement. If you have any difficulty, see the laboratory
demonstrator.
5. Now rotate the prism through 180◦ , repeat the above measurements,
and average the results.
The difference between the vernier readings from each side is twice the
angle of minimum deviation.
8,
6. Repeat using the Hydrogen Geissler tube and the four visible lines of
the Hydrogen spectrum (red, blue and two violet).
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7. Repeat using the Mercury Geissler tube and as many as possible of the
following lines of the Hg spectrum: two close yellow lines separated by
20Å; two greens; a blue–green; and a blue (this is a triplet — use the
center of the image).
8. Repeat using the Helium Geissler tube and any 4 visible lines.
Inlab Tasks
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IT1: Fill in values of δmin in Table 8.2 to see that they are changing monotonically.
8.3.3
Analysis
Plotting the Dispersion Data
1. Using the angles of minimum deviation for each wavelength measured,
along with their uncertainties, and the average value of the prism angle
α, along with its uncertainty calculated above, determine the index of
refraction at each wavelength, along with the uncertainties, and place
them in Table 8.3.
2. Plot a graph of n vs λ, with error bars.
52
Exercise on Dispersion of Glass
3. Save the spreadsheet for future exercises.
20
08
If your graph does not suggest a smooth, monotonically decreasing function,
identify any data points which seem particularly out of place. Bad data will
make your results meaningless.
Postlab Tasks
T1: Print a copy of your data table and your graph and hand it in along
with the question answers.
Recap
8,
8.4
By the end of this exercise, you should know how to calculate the wavelength
from the angle of minimum deviation.
Summary
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8.5
Item
Pre-lab Questions
In-lab Questions
Post-lab Questions
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Pre-lab Tasks
In-lab Tasks
Post-lab Tasks
Number
3
0
0
0
1
1
Received
weight (%)
30
0
0
0
40
30
8.6 Template
Template
My name:
My student number:
My partner’s name:
My other partner’s name:
Today’s date:
Trial #
Left Reading
(◦ )
Right Reading
(◦ )
Prism Angle α
(◦ )
8,
1
2
3
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08
8.6
53
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Table 8.1: Calculation of Prism Angle
54
Exercise on Dispersion of Glass
5893
Hg (y)
5790
Hg (y)
5770
Hg (g)
5461
Hg (g)
5015
H (b-g)
4861
Hg (b-v)
4358
H (v)
4300
H (v)
4101
Hg (v)
4077
Hg (v)
4047
He (y)
?
Right Dev.
(◦ )
δmin
(◦ )
20
08
Na (y)
Left Dev.
(◦ )
8,
H (r)
Wavelength
Å
6563
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Colour
He (g)
?
He (b-g)
?
He (b-v)
?
Table 8.2: n vs Angle of minimum deviation
b–blue; g–green; r–red; v–violet; y–yellow (colours are approximate.)
8.6 Template
5893
Hg (y)
5790
Hg (y)
5770
Hg (g)
5461
Hg (g)
5015
H (b-g)
4861
Hg (b-v)
4358
H (v)
4300
H (v)
Hg (v)
Hg (v)
n
∆n
4101
4077
4047
?
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He (y)
∆δmin
(◦ )
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Na (y)
δmin
(◦ )
8,
H (r)
Wavelength
Å
6563
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Colour
55
He (g)
?
He (b-g)
?
He (b-v)
?
Table 8.3: n vs Wavelength
b–blue; g–green; r–red; v–violet; y–yellow (colours are approximate.)
Exercise on Dispersion of Glass
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56
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Chapter 9
9.1
Purpose
8,
Exercise on the Michelson
Interferometer
9.2
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The purpose of this lab is to use the Micheleson interferometer to detect
small differences in index of refraction.
Theory
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In the Michelson interferometer, a beam of light is split in two and later
recombined. An interference pattern is produced when the two beams recombine. The pattern will be affected if there is a difference in the optical
path length of the two arms of the interferometer. If an air cell is placed
in one of the arms, then the path length in that arm will be different than
if there were a vacuum in place of the air cell, since the index of refraction
for air is not the same as for a vacuum. If we observe the interference pattern while air is removed from the air cell, then the change can be used to
determine the difference between the indices of refraction of air and vacuum.
Due to the air being withdrawn, the total change in optical path length is
∆L = 2(n − 1)t
(9.1)
where n is the index of refraction of air at the wavelength used, and t is the
thickness of the air cell. This change in optical path length may also be given
58
Exercise on the Michelson Interferometer
by
∆L = λ∆m
(9.2)
20
08
since when the total optical path difference is an odd multiple of λ/2, constructive interference occurs and when the optical path difference is an even
multiple of λ/2, destructive interference occurs.
Thus, when t, ∆m, and λ are known, n − 1 is easily calculated.
9.3
Procedure
9.3.1
Apparatus
8,
The following equipment is required for this experiment:
• vacuum pump, Michelson Interferometer with air cell
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• Na lamp, Hg lamp, laser with beam spreader
9.3.2
Preparation
Pre-lab Tasks
PT1: Read the section on the Michelson interferometer on pages 408 and
409 of your text.
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Pre-Lab Questions
PQ1: Combine and rearrange Equations 9.1 and 9.2 to determine how a
linear graph of ∆m vs. λ1 may be produced, and show how n − 1 may be
derived from the graph.
9.3.3
Experimentation
1. Using green light from the Hg lamp (λ = 5461Å), produce circular
interference fringes by adjusting the planes of the fixed mirrors. The
air cell should be in place in one of the arms. Use a vacuum pump to
withdraw the air from the cell.
9.3 Procedure
59
2. Count the number of fringes, ∆m that appear or disappear. It may be
easier to determine ∆m while air is admitted into the cell.
3. Repeat this several times and average the results for consistency.
5. Measure the inside thickness of the air cell.
8,
Inlab Questions
20
08
4. Repeat the above procedure using the Na lamp, and any other source
possible. Some possibilities are the Mercury light (with the approriate
filter) for the yellow lines at 5770 Åand 5790 Å, the Mercury light (with
the approriate filter) for the blue line at 4358 Å.
IQ1: Did the number of fringes change monotonically with the wavelength
of light? Did it change in the direction expected?
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Inlab Tasks
IT1: Summarize your results to the lab instructor.
9.3.4
Analysis
1. Plot the linearized graph, and do a least squares fit.
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Postlab Discussion Questions
Q1: Calculate the index of refraction of air from the results of the least
squares fit, and compare this with the expected value. Does the y-intercept
of your graph match your expectations? Explain.
Postlab Tasks
T1: Hand in the question answers, along with the graph and your calculations.
60
9.4
Exercise on the Michelson Interferometer
Recap
Summary
Item
Pre-lab Questions
In-lab Questions
Post-lab Questions
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Pre-lab Tasks
In-lab Tasks
Post-lab Tasks
Number
1
1
1
1
1
1
Received
8,
9.5
20
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By the end of this exercise, you should know how to use a Michelson interferometer.
weight (%)
10
20
10
10
40
10
9.6 Template
9.6
61
Template
number of fringes
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8,
wavelength
(Å)
20
08
My name:
My student number:
My partner’s name:
My other partner’s name:
Today’s date:
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Table 9.1: Mercury lamp data
Exercise on the Michelson Interferometer
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8,
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62
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Chapter 10
Exercise on Newton’s Rings
Purpose
8,
10.1
The purpose of this lab is to investigate interference in thin films.
Theory
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10.2
Various methods of producing interference fringes can be classified into two
types:
1. division of a wavefront into two or more beams (double slit, Lloyd
mirror, Fresnel biprism) which are later recombined, and
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2. division of the amplitude of an extended portion of the wavefront into
beams which are later recombined
Unlike the division of the wavefront, thin film interference does not require
a narrow line source or meticulous alignment of auxiliary equipment. The
familiar colours in soap films and oil slicks are excellent examples of thin film
interference.
When light is incident on a thin film, part of it is reflected from the first
surface, and part is reflected from the second surface. These recombine at
the first surface to form the reflected beam, and the rest combine to form
the transmitted beam at the second surface. Because of the optical path difference between the two reflected beams, a phase difference exists between
64
Exercise on Newton’s Rings
10.2.1
8,
20
08
these beams, leading to interference effects. In addition, there are phase differences introduced because of reflections. Whenever reflection occurs from
an optically denser medium, a phase change of π is introduced. Thus, when
the total optical path difference is an odd multiple of λ/2, constructive interference occurs. When the optical path difference is an even multiple of
λ/2, destructive interference occurs. These effects result in the formation of
interference fringes as observed with the reflected light. Fringes also occur in
the transmitted beam, except that these are the complement of the reflected
fringes, and are much harder to observe.
The fringes are essentially constant thickness contours of the film. They
can be used to map the irregularities in the film thickness and/or determine the film thickness. The Michelson Interferometer and the Fabry-Perot
Interferometer are good examples of devices operating on this principle.
Newton’s Rings
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Newton’s Rings form a circular interference pattern due to the reflections
occurring in an air wedge, formed around the point of contact between a
spherical surface and a plane surface, as shown in Figure 10.1.
Let rm be the radius of the mth dark ring. A is the center of curvature,
B is the point of contact, and R is the radius of the spherical surface of
the plano-convex lens. Note that m is an integer, and h is the height of the
surface of the lens above the plane surface at radius rm .
From the geometry, it can be shown that
2
rm
= 2Rh − h2
(10.1)
for this setup.
Then mλ = 2h, and a dark fringe is produced at the center (because of
the phase change due to reflection). Since h R, this means
2
rm
≈ 2Rh = Rmλ
If rm is measured as a function of m for different values of λ, and a graph of
2
rm
vs mλ is plotted, a straight line with slope R should be obtained.
65
20
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10.2 Theory
8,
A
R
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R
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B
yXX
X
h
rm
Figure 10.1: Plano–Convex Lens
:
Exercise on Newton’s Rings
20
08
66
Observer
8,
Travelling Microscope
Parallel Light
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Thin Glass Plate
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Plano-Convex Lens
Optical Flat
Figure 10.2: Newton’s Rings Setup
10.3 Procedure
10.3
Procedure
10.3.1
Preparation
67
20
08
Pre-lab Tasks
PT1: Read the section on the Newton’s rings on pages 406 and 407 of your
text.
Pre-Lab Questions
PQ1: Derive Equation 10.1.
Apparatus
8,
10.3.2
The following equipment is required for this experiment:
• planoconvex lens, glass plates, lens holders
10.3.3
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• Na lamp
Experimentation
Measurement of Fringe Diameters
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Figure 10.2 shows the experimental arrangement to be used. Light from the
Na arc lamp (λ = 5893Å) is reflected to the flat surface of the plano-convex
lens by a thin glass plate inclined at 45◦ to the vertical. Light is reflected at
the boundaries of the air film between the lens and the optical flat. These two
reflections pass through the glass plate into the travelling telescope, which
magnifies the rings and can be used to measure their diameters.
1. Carefully adjust the telescope so that the center of the cross-hairs pass
through the center of the fringe pattern (travelling at right angles to
the fringes).
2. Set the cross-hairs tangentially on each dark ring, beginning at the
10th ring on one side, and moving over to the 10th ring on the other
side. Record the positions of each dark ring as a function of m. The
diameters of each ring can be found from these measurements.
68
Exercise on Newton’s Rings
2
3. If the order of the central dark spot is m = 0, make a table of rm
vs m.
20
08
4. After measuring the fringes, replace the lens with a reticule and measure a wide spacing on it so that you can determine the calibration for
the microscope.
Inlab Questions
IQ1: Using the calibration of the reticule, determine the diameter of the
biggest ring which you measured. Does the value fit with what you can
observe by eye?
8,
Inlab Tasks
IT1: Summarize your results to the lab instructor.
Analysis
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10.3.4
1. Determine the slope and y-intercept of the line passing through your
data points.
Postlab Discussion Questions
Q1: Calculate the value of R. Does it seem reasonable?
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Q2:Does the y-intercept of your graph make sense? Explain.
Postlab Tasks
T1: Hand in the question answers, along with the graph and your calculations.
10.4
Recap
By the end of this exercise, you should know how to use and calibrate a
micrometer scale.
10.5 Summary
Summary
Item
Pre-lab Questions
In-lab Questions
Post-lab Questions
Received
1
1
1
weight (%)
10
20
20
10
20
20
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8,
Pre-lab Tasks
In-lab Tasks
Post-lab Tasks
Number
1
1
2
20
08
10.5
69
70
Template
My name:
My student number:
My partner’s name:
My other partner’s name:
Today’s date:
fringe number
1
20
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10.6
Exercise on Newton’s Rings
reading reading
(left) (right)
3
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4
8,
2
5
6
7
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9
10
Table 10.1: ?
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Chapter 11
11.1
Purpose
8,
Exercise on Generalized Least
Squares Fitting
11.2
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The purpose of this exercise is to show you how to do least sqaures fits to
polynomial functions in Excel TM .
Introduction
Previously you have done curve fitting in two dimensions. Now you will learn
how to extend that to multiple dimensions.
Theory
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11.3
11.3.1
Non-linear
Linearizable
Some equations, such as
y = Ae(Bx)
can be treated fairly simply. Linearize and do a linear least squares fit, as
you have done in the past. (Note: “Least Squares” applies to transformed
quantities, not original ones so gives a different answer than you would get
from a least squares fit in the untransformed quantities; remember in general
72
Exercise on Generalized Least Squares Fitting
20
08
the idea of a line of “best fit” is not unique. For example, for the equation
shown, ln y vs. x is linear, so you can do a least squares fit in ln y, but this
will not give the same result as a least squares fit in y, since the sum of
squares of ln y will depend on the data in a different way than the sum of
squares in y.)
Non-linearizable
These will be dealt with in “Exercise on the Solver Spreadsheet Feature”.
11.3.2
Linear
Some equations, such as this
8,
General
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y = b0 + b1 x 1 + b2 x 2 + · · · bk x k
are linear, although in multiple variables. We can create a matrix of independent data



X=

x11 x12
x21 x22
..
..
.
.
xn1 xn2
. . . x1k
. . . x2k
. . . ..
.
. . . xnk





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from the x values, where xij means variable xj for data point i and form a
vector of dependent data


y1


Y =  ... 
yn
where yi is the y data for data point i.
This creates a system which can be solved using functions in a spreadsheet.
11.3 Theory
73
Polynomial
Consider an equation such as this:
y = b0 + b1 x + b2 x 2 + · · · bk x k
20
08
This is just a special case of the previous situation above, eg. x1 = x, x2 = x2 ,
x3 = x3 , etc. (or x1 = 1/x, x2 = 1/x2 , etc.)
What about fit with skipped orders?
eg. y = a + b/x2 + c/x5
In this case, x1 = 1/x2 , x2 = 1/x5 .
11.3.3
Goodness of fit
8,
A linear or polynomial least squares fit does not need starting values for
parameters like the solver function would need.
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Often you must choose between different fits because you do not know what
type of equation to use. In this case you want to be able to answer the
question “Which fit is better?”
1. If both fits have the same number of parameters, then the better fit is
the one with the smaller SSE in the same quantity. (In other words,
if you’re comparing a fit in y vs. x to one in ln y vs. x, you will first
have to calculate the SSE of both in y vs. x. If you have linearized
an equation to calculate a fit, you can still use that fit to calculate the
SSE in the original quantity afterward.)
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2. One or both of the fits may have some parameters which are not “statistically significant”; (i.e. lots of parameters close to 0 are probably
meaningless.) How close to 0 is “close enough”?
• RULE: Adding more parameters → smaller SSE, (however a small
change in SSE may not be significant.) Whether or not the added
parameters are significant can be determined statistically if the fit
is a linear one or one which can be linearized.
To actually determine whether the change is big enough, proceed as
follows:
74
Exercise on Generalized Least Squares Fitting
(a) Do fit with g + 1 parameters (as above); calculate the sum of
squares error and call it SSE1 . If you use regression from a
spreadsheet, you can determine SSE from the results. Remember
SSE1 = s1 2 ν1 ; in this case ν1 = n − (g + 1).
(c) Calculate s3 as follows:
s
s3 =
and let ν3 = k − g.
(d) Calculate F as follows:
20
08
(b) Do fit with k+1 parameters (as above); calculate SSE2 . As above,
SSE2 = s2 2 ν2 and in this case ν2 = n − (k + 1).
SSE1 − SSE2
k−g
s3 2
s2 2
If F is big, then include the extra parameters. (In this case, it
means the SSE changed a lot by adding the extra parameters,
which is what would happen if they were really important.) How
big is “big”?
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8,
F =
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(e) Look up Fα,ν3 ,ν2 from a table of the F distribution in a statistics text1 , where α determines the confidence interval; typically
α = 0.05 for a 95% confidence interval. If the F you calculated
is greater than the table value, then keep the extra parameters.
Note: In the table, you are given quantities ν1 and ν2 ; you should
use your calculated value of ν3 in place of ν1 in the table. Doing
it this way keeps the table in the same form you will find it in a
statistics text.
The following example illustrates how to do this for a linear fit. Usually we
want to compare two fits; in this example, we will compare 3 fits to illustrate
the process more clearly. We will compare 2 fits at a time, and in each case
we will use g + 1 to denote the number of parameters in the “smaller” fit 2
, and k + 1 to denote the number of parameters in the “bigger” fit, so k is
always bigger than g.
1
You can also use the FINV function of a spreadsheet like Excel or OpenOffice.
Why not just g? Because g is the degree of the polynomial, which has g+1 parameters.
For example a polynomial of degree 2, such as Ax2 + Bx + C has 3 parameters, namely
A, B, and C.
2
11.4 Example
11.4
75
Example
y
1
2
4
8
15
25
45
8,
x
100
85
70
50
36
20
10
20
08
Consider the data shown in Table 11.1 and plotted in Figure 11.1. (Error
bars have been omitted for simplicity.)
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Table 11.1: Sample Data
Figure 11.1: Plot of Sample Data
It should be obvious that some possible equations for a fit to this data
may be polynomials in 1/x.
A fit to y = A + B/x gives us Table 11.2.
Exercise on Generalized Least Squares Fitting
1/x
0.0100
0.0118
0.0143
0.0200
0.0278
0.0500
0.1000
y
1
2
4
8
15
25
45
20
08
76
Table 11.2: Data for Fit to A + B/x
488.15
30.56
0.98
255.18
1502
-2.02
1.37
2.43
5
29.43
A
σA
s
ν
SSE
8,
B
σB
R2
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Table 11.3: LINEST 2 parameter output
Figure 11.2: Graph of Fit to A + B/x
11.4 Example
77
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8,
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08
Notice that the curve cannot “bend enough”, and so we will see what
happens if we add another parameter. To do this we simply add another
column to the data, and change LINEST so that it now points to the new
block of x data, which includes two columns.
Exercise on Generalized Least Squares Fitting
1/x
0.01000
0.01176
0.01429
0.02000
0.02778
0.05000
0.10000
1/x2
0.000100
0.000138
0.000204
0.000400
0.000772
0.002500
0.010000
y
1
2
4
8
15
25
45
20
08
78
Table 11.4: Data for Fit to A + B/x + C/x2
784.5
62.79
1.04
4
4.31
-6.45
1.09
#N/A
#N/A
#N/A
8,
-2713.14
562.24
1
707.95
1527.11
A
σA
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C
σC
R2
Table 11.5: LINEST 3 parameter output
Note how the parameters are rearranged when there are more of them.
(The parameters are actually arranged on the top row of the output in reverse
order of the data columns.)
It’s now possible to compare the two fits to see if the third parameter
makes a big enough change to keep. To do that, we do the following:
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1. Get s for the fit with more parameters. (We’ll refer to this as the “big”
fit.) Call this s2 . In our example, that would be s2 = 1.04.
2. Calculate s3 due to the difference of the fits. Since we’re only checking
for the effect of one parameter,
s
r
SSE small − SSE big
29.43 − 4.31 √
s3 =
=
= 25.12 = 5.01
νsmall − νbig
1
3. Calculate F =
s3 2
s2 2
=
5.012
1.042
= 23.2
11.4 Example
79
4. Look up the value of F0.05,νsmall −νbig ,νbig = F0.05,1,4 or calculate it using
the FINV function of a spreadsheet. (The values go in the order given.)
From Table 11.9, F0.05,1,4 = 7.71 is the value to compare.
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5. Since our calculated value for F is bigger than the value from the table,
that means our extra parameter made a big difference, so we should
keep it.
Exercise on Generalized Least Squares Fitting
8,
20
08
80
Figure 11.3: Graph of Fit to A + B/x + C/x2
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This process can be repeated as many times as we wish.
11.4 Example
81
1/x2
0.000100
0.000138
0.000204
0.000400
0.000772
0.002500
0.010000
1/x3
0.00000100
0.00000163
0.00000292
0.00000800
0.00002143
0.00012500
0.00100000
y
1
2
4
8
15
25
45
20
08
1/x
0.01000
0.01176
0.01429
0.02000
0.02778
0.05000
0.10000
Table 11.6: Data for Fit to A + B/x + C/x2 + D/x3
-12121.37
4672.18
0.78
3
1.83
1156.6
189.98
#N/A
#N/A
#N/A
-9.93
1.91
#N/A
#N/A
#N/A
A
σA
8,
60476.14
29909.81
1
837.72
1529.6
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D
σD
R2
Table 11.7: LINEST 4 parameter output
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It is not immediately obvious which of the above curves fits the data
“best”. We could even go on adding higher and higher powers of 1/x until
we had no more degrees of freedom 3 left, but once we get no significant
change, it’s time to stop.
Table 11.8 summarizes the results of the various fits.
3
The number of degrees of freedom in a fit is the number of data points beyond the
bare minimum for that fit. So, for an average it is n − 1, since only one value is needed;
for a straight line it is n − 2, since two points are needed, etc. In general,
ν =n−m
where m is the number of parameters in the fit to be determined. Note that when you
have no degrees of freedom, you have no idea of the “goodness” of your data, and thus
cannot determine the standard deviation. Once you have even one degree of freedom, you
can do so.
Quantity
Fit
A + B/x + C/x2 A + B/x + C/x2 + D/x3
1.04
0.78
4
3
4.31
1.83
-6.45
-9.93
1.09
1.91
784.5
1156.6
62.79
189.98
-2713.14
-12121.37
562.24
4672.18
60476.14
29909.81
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s
ν
SSE
A
σA
B
σB
C
σC
D
σD
A + B/x
2.43
5
29.43
-2.02
1.37
488.15
30.56
20
08
Exercise on Generalized Least Squares Fitting
8,
82
Table 11.8: Comparisson of Fit Parameters
• Note that the SSE gets smaller as the number of parameters increases,
but the change gets smaller.
• Note also that when a parameter is added, all of the previous parameters change as well.
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• Even though it is considered “insignificant”, the D parameter is bigger
than all of the rest! (However, note the size of its standard error.
Remember also that it gets divided by x3 , which will range in size from
1000 → 1000000.)
83
8,
20
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11.5 Procedure
Figure 11.4: Graph of Fit to A + B/x + C/x2 + D/x3
Procedure
11.5.1
Investigation
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11.5
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The following functions are polynomials in λ1 or something similar. Set up
the appropriate tables for each and use LINEST to do a least squares fit for
each.
Then use the goodness of fit criteria to determine the best.
n=A+
B
λ
(11.1)
n=A+
B
λ2
(11.2)
n=A+
B
C
+ 2
λ
λ
(11.3)
n=A+
B
C
+ 4
2
λ
λ
(11.4)
84
Exercise on Generalized Least Squares Fitting
\ ν1
1
2
3
4
5
6
..
.
1
161.4
18.51
10.13
7.71
6.61
5.99
..
.
2
199.5
19
9.55
6.94
5.79
5.14
..
.
3
215.7
19.16
9.28
6.59
5.41
4.76
..
.
4
224.6
19.25
9.12
6.39
5.19
4.53
..
.
5
230.2
19.3
9.01
6.26
5.05
4.39
..
.
6
234
19.33
8.94
6.16
4.95
4.28
..
.
···
···
···
···
···
···
···
...
∞
254.3
19.5
8.53
5.63
4.36
3.67
..
.
∞
3.84
3
2.6
2.37
2.21
2.1
···
1
20
08
ν2
Table 11.9: F–Distribution Table (α = 0.05)
8,
Using LINEST for the first two parameter fit
• Put the function in C20, with constant=1 and stats=1.
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• Highlight C20 to D24.
• Press hF2i followed by hCTRLihSHIFTihENTERi.
• Some of the important parameter names have been indicated.
For the second two parameter fit, you can just copy and paste as needed
(or adapt the steps above for the new location.)
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Using LINEST for the three parameter fits
It works in a similar way, but instead of C20 to D24, highlight C20 to E24.
(There is one more parameter so another column is needed.) Note the way
the parameters are arranged in this case.
Inlab Tasks
IT1: Do each of the fits above and fill in Table 11.10.
IT2: Use the F test to determine whether fit 11.3 is better than fit 11.1 or
fit 11.2.
11.6 Summary
85
Inlab Questions
IQ1: Based on the SSE values, which was better; fit 11.1 or fit 11.2?
11.5.2
20
08
IQ2: Based on the SSE values, which was better; fit 11.3 or fit 11.4?
Analysis
Postlab Discussion Questions
Q1: After you have completed the “Exercise on the Solver Spreadsheet
Feature”, what is the best fit for the data?
Summary
Number
0
2
1
Received
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Item
Pre-lab Questions
In-lab Questions
Post-lab Questions
8,
11.6
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Pre-lab Tasks
In-lab Tasks
Post-lab Tasks
0
2
0
weight (%)
0
40
20
0
40
0
86
Template
My name:
My student number:
My partner’s name:
My other partner’s name:
Today’s date:
Fit
11.1
11.2
11.3
11.4
A
σA
B
σB
20
08
11.7
Exercise on Generalized Least Squares Fitting
C
12.1
8,
From Solver
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Table 11.10: Fit comparison
σC
SSE
20
08
Chapter 12
12.1
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8,
Exercise on Sketching
Functions and Extracting
Parameters
Purpose
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The purpose of this exercise is to show you how to determine starting values
for the tools like the Solver add-in in Excel TM .
12.2
Introduction
Previously you learned how to linearize functions and then use least squares
fitting to get parameters from the graph. Another exercise explains how
to use spreadsheet features to do least squares fitting with non-linearized
functions. However, to do that it is necessary to have reasonable, i.e. order
of magnitude, starting values for the parameters involved. Now you are going
to learn how to do that.
88
Exercise on Sketching Functions and Extracting Parameters
12.3
20
08
As long as you have a ballpark figure for each parameter, you will be able to
use mathematical tools to refine the values. But without knowing whether
a parameter has a value closer to 0.00005, 30, or 7 × 109 , the mathematical
tools will be pretty useless. The techniques discussed here are to give you
order of magnitude values; in other words, values that are about right to one
significant digit. That should be close enough for the mathematical tools to
work.
Theory
8,
In first year calculus, you should have learned about sketching functions.
This will be discussed now because of the way that parameters for functions
can be extracted from plots of observed data; in other words, knowing how
certain parameters affect the graph of a function allows one to determine
those parameters from the graph of a function.
Critical Points
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12.3.1
A critical point of a function occurs where the slope is zero. In other words,
df (x)
=0
dx
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If a graph of actual data has a critical point, then this can be used according
to the equation above. (Critical points also occur where the function is
undefined. These may be observable as well.)
12.3.2
Limits and Asymptotes
The limits of a function as x → ∞ or any other point will often be useful if
either of the following is true:
• function has a limit; in this case, if we can observe values of the function
near enough to the limit, we can approximate the limit. Keep in mind
that for one function, there may be useful limits as you approach a
variety of x values, so you may need to do this several times to extract
as much information as possible.
12.3 Theory
89
• function has an asymptote; an asymptote is just a special kind of limit.
In this case, for a function f (x), an asymptote exists if
lim f (x) − (mx + b) = 0
x→∞
1. Calculate m by using the fact that
df (x)
=m
x→∞ dx
lim
20
08
for some choice of m and b. If they exist, then m and b can be found
as follows:
8,
ie. See if you can observe the data approaching an asymptote. If
so, then you can draw in the line and calculate the slope m.
2. Calculate b according to the equation above, using this value of
m. In other words,
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lim f (x) − (mx + b) = 0
x→∞
thus
lim f (x) − mx − b = 0
x→∞
and so
lim f (x) − mx = b
x→∞
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b
ie. extrapolate the asymptote you have drawn to where it crosses
the y axis and determine b. (You don’t actually need to do this;
given any two points on a line you can calculate both m and b.)
Thus if asymptotes exist, they can be used to determine parameters of
the original function.
12.3.3
Roots
If a function has zeroes, i.e. points where f (x) = 0, then these can also be
useful if they can be observed in the data. (For that matter, if the function
takes on any fixed value then this can be used.)
90
Exercise on Sketching Functions and Extracting Parameters
12.3.4
Modified Functions
12.3.5
20
08
At times, some of the above rules will be useful if applied to a modified version
of the function; for instance f (x) may not have an asymptote, but ln (f (x))
might. With practice you should get better at spotting these situations.
Substitution
12.4
Procedure
12.4.1
Preparation
Pre-Lab Tasks
8,
If all but one of the parameters for a function can be estimated by using the
previous methods, then the final parameter can be estimated by substituting
the other values into the equation for a single data point.
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PT1: Using the data from the spreadsheet given on the website, plot an
x − −y graph of n versus λ and bring it to the lab.
12.4.2
Investigation
The function shown in Equation 12.1 is non-linear. Analyze it for roots,
asymptotes, etc. as discussed above to determine how to get estimates for
A, B, and C from the graph of this function.
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b
B
(12.1)
λ−C
The functions shown in Equation 12.2 and 12.3 are also non-linear.
n=A+
I=
I=
A cos (Bθ + C) + D
Acos2 (Bθ + C) + D
(12.2)
(12.3)
Analyze each function for roots, asymptotes, etc. as discussed above to
determine how to get estimates for A, B, C, and D from the graphs of these
functions.
91
8,
20
08
12.4 Procedure
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n
Fe
b
λ
Figure 12.1: Dispersion
Exercise on Sketching Functions and Extracting Parameters
8,
20
08
92
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I
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b
θ
Figure 12.2: Malus’ law
12.5 Summary
93
Inlab Tasks
IT1: On Figure 12.1, show how approximate values for the parameters can
be determined from the graph.
20
08
IT2: On Figure 12.2, show how approximate values for the parameters can
be determined from the graph. (For the parameters that will be different for
each function, highlight them in some way to distinguish the two functions.)
Write your answers in Table 12.1.
Inlab Questions
12.4.3
Analysis
8,
IQ1: Using the graph from the spreadsheet in the pre-lab question, what
are your estimates for A, B, and C? What sort of range do you think is
reasonable in each of the estimated values?
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Postlab Questions
This will only be able to be answered after you have done the appropriate
exercises.
Q1: When you have data values from “Exercise on Dispersion of Glass” and
“Exercise on Malus’ Law” determine approximate values for the parameters.
Summary
Item
Pre-lab Questions
In-lab Questions
Post-lab Questions
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12.5
Pre-lab Tasks
In-lab Tasks
Post-lab Tasks
Number
0
1
1
1
2
0
Received
weight (%)
0
20
20
10
50
0
94
12.6
Exercise on Sketching Functions and Extracting Parameters
Template
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8,
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My name:
My student number:
My partner’s name:
My other partner’s name:
Today’s date:
95
parameter
how to get it
Equation 12.1
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Equation 12.2
20
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12.6 Template
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Equation 12.3
Table 12.1: Extracting parameters
Exercise on Sketching Functions and Extracting Parameters
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8,
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96
20
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Chapter 13
13.1
Purpose
8,
Exercise on the Solver
Spreadsheet Feature
13.2
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The purpose of this exercise is to introduce you to the Solver add-in tool in
Excel TM .
Introduction
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One of the purposes of labs is to develop the ability to analyze data. The
spreadsheet is one of the most powerful and ubiquitous tools available for
this purpose. One very useful feature is described below. The purpose of
this exercise is to introduce you to it.
13.3
Theory
At times you will wish to find a maximum or minimum value for a function
over a given range of input values. For instance, given
y = f (x)
one might need to know what value of x maximizes the value of y. If f (x)
is monotonically increasing or decreasing, then the function will have its
maximum value at one of the endpoints of the interval in question. In other
98
Exercise on the Solver Spreadsheet Feature
cases, it may be much harder to determine. If the function is continuous,
then the function will have a local maximum or minimum wherever
y0 = 0
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These points can be approximated in a similar way to the way roots of a
function are found, as just described. In this case, if we use successive values
of the function to estimate both the first AND second derivatives of the
function at a point, then it is possible to estimate the point at which the
slope will be zero.
There are many numerical algorithms for finding good numerical approximations to these solutions, and Excel Pro has one of these built in.
Although there may be many possible candidates for the solution to the
above equation, the spreadsheet will only check one such candidate, and the
one checked will depend on the starting value given. The starting value
is a value for x which should be somewhere in the vicinity of the candidate
being approximated. The caveat is, the more you know about the expected
solution(s), the easier it is (they are) to find. (In practice, most functions
are sufficiently well-behaved that there may be at most a handful of possible
candidates, and usually only one. In these cases, such algorithms will usually
do a good job of approximating them.)
To find the x value which maximizes y value, we must optimize the
equation
y = f (x)
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That is, we must “find the value of x which gives the maximum (or minimum)
value for y”.
A least squares fit is really just a kind of optimization; it amounts to
finding the values of m and b (for a linear fit) which minimize the sum of
squares error.
In this case, the function to be minimized is
S = S(m, b)
This is a function of 2 variables, and we want to find the values of both m
and b which minimize S.
For a non-linear fit, then the equation for S and thus the parameters
involved would be different.
13.3 Theory
13.3.1
99
Uncertainties in Parameters
20
08
One difficulty with using these methods for fitting is that we cannot easily
generate standard errors in the fit parameters. While there is no general
way to do this, we can get some idea of the sensitivity of the parameters to
changes in data values by varying the data values within their experimental
uncertainties. This can be done for a function with parameters A and B as
follows:
1. Replace the variable values in the data so that, for instance,
xi 0 = xi + 2(rand() − 0.5)(∆xi )
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8,
where the xi are the given values with uncertainties in them. (The same
can be done for y values.) (The rand() function of the spreadsheet
gives a random value between 0 and 1, so subtracting 0.5 from it gives
a random value between −0.5 and +0.5, and multiplying this by 2 gives
a value between −1 and 1 ; thus we get a value between xi − ∆xi and
xi + ∆xi .)
2. Now you will see that any time you recalculate, all of those values (and
thus the results) will change as well.
3. Copy these xi 0 and xi 0 values as values to another part of the spreadsheet.
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4. Change your formula cell to point to these new copied xi 0 and xi 0 values
instead of the previous xi and yi . Now the the solution A and B values
will depend on the uncertainties in x and y.
5. To see the effect on the parameters calculated, you will need to re-run
the solver.
6. Do this 5 or 10 times and use the ranges in the parameter values as
estimates for the uncertainties in the parameters.
(You might wonder why we don’t just change all of the data values to
their values with uncertainties added, and then see how much that changes
the answer, thus giving us the maximum uncertainty. The problem is that in
complicated calculations, the effects may cancel each other out, so there is no
apparent uncertainty. If we were subtracting the numbers, this is obvious.)
100
Exercise on the Solver Spreadsheet Feature
yi
ŷi = Axi 2 + B
0.00
0.28
2.00
1.46
2.70
3.32
4.30
4.50
P
S=
(ŷi − yi )2
xi
0.40
0.77
1.35
1.72
(ŷi − yi )2
0.08
0.29
0.38
0.04
20
08
i
1
2
3
4
0.79
Table 13.1: Least Squares Fit data
13.3.2
Solver Example: Least Squares Fitting
Getting the parameter values
8,
Consider the data in Table 13.1.
In this case, we want to do a non-linear fit. Specifically, we wish to
minimize the value of S where
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S = S(A, B)
where
S=
X
(ŷi − yi )2
At first glance, S does not seem to depend on A and B. However, if we
expand the above equation we get
X
2
S=
Axi 2 + B − yi
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since
ŷi = Axi 2 + B
Note that if we change A and /or B, then the values of ŷ will change and
thus S will change also.
If we set up a spreadsheet like Table 13.1, then one can vary A and
B manually and observe the changes in S. Having done so, it is easy to
automate the process using the solver in the spreadsheet.
Note also that we could have used any function in place of the equation
y = Ax2 + B in the fourth column above, and the calculation of S would
remain the same. Thus it now becomes easy to do least squares fits of nonlinear functions as well as linear ones.
13.3 Theory
original xi
0.40
0.77
1.35
1.72
∆xi
0.01
0.02
0.03
0.05
original yi
0.00
2.00
2.70
4.30
∆yi
0.05
0.05
0.05
0.05
xi 0
0.39
0.76
1.36
1.7
yi 0
0.04
2.03
2.71
4.34
20
08
i
1
2
3
4
101
Table 13.2: Finding parameter sensitivities
1. Set up a cell containing the formula
X
(ŷi − yi )2
8,
where ŷi , and yi , are references to variable cells.
2. Set
with A and B as your variables, and the cell containing
P up the solver
2
(ŷi − yi ) as your solution cell. Configure the solver to minimize this.
13.3.3
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3. Run the solver a few times to see if your results converge.
Getting the parameter uncertainties
To get some idea of the sensitivity of the parameters to changes in data
values by varying the data within its experimental uncertainty, proceed as
follows. (Note in this case the variables are the actual data point coordinates
themselves.)
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1. Make new columns of data so that, for instance,
xi 0 = xi + 2(rand() − 0.5)(∆xi )
and
yi 0 = yi + 2(rand() − 0.5)(∆y i )
as shown in Table 13.2.
2. Now you will see that any time you recalculate, all of those values will
change as well. Copy these xi 0 and yi 0 values and use the Paste Special
command to paste the values as numbers, not formulas into somewhere
in the spreadsheet.
102
Exercise on the Solver Spreadsheet Feature
3. Change your formulae for (ŷi − yi )2 so that it is these new values for
yi 0 are used instead of the yi values from before.
20
08
If you don’t point to the pasted values, then the random variation will be
happening while the solver is running, which could make your results very
unpredictable.
4. Run the solver. Now the SSE will depend on the uncertainties in x and
y.
5. To see the effect of the uncertainties on the parameters calculated, you
will need to repeat the cut-and-paste and then re-run the solver.
13.3.4
8,
6. Do this 5 or 10 times and use the ranges in the parameter values as
estimates for the uncertainties in the parameters.
Solver in Excel
TM
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Solver is an add-in tool in Excel, which is simple to use. To be able to use
it, do the following (in Excel 2007):
1. Click on the “Orb”; (i.e. the office icon)
2. Click on the “Excel options” button at the bottom of the dialog.
3. Select “add-ins”.
4. Select “Solver add-in”.
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It will then show up under “Analysis” on the “Data” tab.
When you invoke it, you will be presented with a dialogue box as shown
in Figure 13.1.
Set Target Cell In our example of a least squares fit, this would be the
SSE
13.3 Theory
103
SSEmin -
20
08
:
A,B,
C, etc.
8,
Figure 13.1: Solver parameters
• min
• max
• value
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Equal To There are three choices here:
In the case of the least squares fit, we would like to minimize the SSE.
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By Changing Cells All of the parameters of our fit need to be changed
in order to find the solution.
Buttons
Solve
Options There are some options you can choose, which are indicated
by a dialogue box as shown in Figure 13.2.
Max Time
Exercise on the Solver Spreadsheet Feature
8,
20
08
104
Figure 13.2: Solver options
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Iterations
Precision
Tolerance (%)
Buttons
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OK There are other options about the type of algorithm used; usually
the default is fine. You can try others if you wish.
When the solver is finished, you should see a dialogue box as shown in
Figure 13.3.
Buttons
Keep Solver Solution
105
Figure 13.3: Solver results
8,
Restore Original Values
OK
13.4.1
13.4.2
Procedure
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13.4
20
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13.4 Procedure
Preparation
Investigation
For our experiment, we want to refine parameters for the following equation:
B
(13.1)
λ−C
1. Follow the procedure of section 13.3.2 above, sustituting n for y, λ for x
and equation 13.1 instead of Axi 2 +B. Use the values from “Exercise on
Sketching Functions and Extracting Parameters” for the starting values
of A,B, and C. Use the data from the spreadsheet provided unless you
have already completed the “Exercise on Dispersion of Glass”.
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n=A+
2. Run the solver to determine optimal values for A, B, and C.
Inlab Questions
IQ1: What values did you get for A, B, and C? Were they in the same
ballpark as your starting values? What values of SSE did you get?
106
Exercise on the Solver Spreadsheet Feature
Inlab Tasks
IT1: Demonstrate you process and results to the lab supervisor.
Analysis
20
08
13.4.3
1. Follow the procedure of section 13.3.3 above, to determine uncertainties
in A, B, and C. Fill the values in Table 13.4. (You can use an uncertainty of 10 Angstroms for λ and an uncertainty of 0.001 for n unless
you have already completed the “Exercise on Dispersion of Glass”.)
Postlab Questions
Postlab Tasks
8,
Q1: Sate the values of A, B, and C with their uncertainties.
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T1: Print off your data table and solver results and hand them in along with
the question answers.
13.5
Recap
By the end of this exercise, you should know how to use the solver function
to optimize functions.
Summary
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13.6
Item
Pre-lab Questions
In-lab Questions
Post-lab Questions
Pre-lab Tasks
In-lab Tasks
Post-lab Tasks
Number
0
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0
1
1
Received
weight (%)
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40
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13.7 Template
13.7
107
Template
Equation
A
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My name:
My student number:
My partner’s name:
My other partner’s name:
Today’s date:
B
C
∆A
B
∆B
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A
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Table 13.3: Fit values
C
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Table 13.4: Parameter uncertainties
∆C
Exercise on the Solver Spreadsheet Feature
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108
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Chapter 14
14.1
Purpose
8,
Exercise on the Goal Seek
Spreadsheet Feature
14.2
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The purpose of this exercise is to introduce you to the Goal Seek tool in
Excel TM .
Introduction
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One of the purposes of labs is to develop the ability to analyze data. The
spreadsheet is one of the most powerful and ubiquitous tools available for
this purpose. One very useful feature is described below.This exercise will
introduce you to it.
14.3
Theory
Often when dealing with numerical data, one will be faced with functions
which are non–invertible. Assume
y = f (x)
If f (x) is invertible, then there exists a function f −1 (x) such that
x = f −1 (y)
110
Exercise on the Goal Seek Spreadsheet Feature
so that given a value y0 , we can calculate the value of x0 which corresponds
to y0 1 in the original function; ie.
y0 = f (x0 )
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In many cases, an inverse function does not exist. This can happen for a
variety of reasons:
1. The function is many–to–one. Many x values give the same y value, so
there is not a unique inverse. (eg. y = sin(x))
2. The function is sufficiently complex that no simple inverse exists. (eg.
y = x + ex ; y 0 > 0 everywhere, but no simple inverse function exists.)
8,
To find the x value which produces a given y value, we must solve the
equation
y0 = f (x)
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That is, we must “find the value of x which makes the equation true”. Solving
this equation is the same as solving the equation
f (x) − y0 = 0
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Various mathematical algorithms exist for finding good numerical approximations to these solutions, and several spreadsheets, including Excel, have
one of these built in.
Although there may be many possible solutions to the above equation, a
tool such as the spreadsheet will only produce one at a time. In fact most
such tools will only give one solution, and the one given will depend on the
starting value used. The starting value is a value for x which should be
somewhere in the vicinity of the solution being approximated. The caveat
is, the more you know about the expected solution(s), the easier it is (they
are) to find.
Consider the function shown in Figure 14.1. The desired solution is the
x–value where the function crosses the x–axis.
One algorithm for solving an equation F (x) = f (x) − y0 = 0 is the
Newton-Rhapson or Newton’s method.2 The algorithm starts with an
1
The value of x for which f (x) = 0 is known as a root of f (x).
Other methods include the bisect method which works if you know one value of x for
which f (x) > 0 and another for which f (x) < 0.
2
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14.3 Theory
F (x1 )
x0
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x2
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?
?
6
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x1
F (x0 )
Figure 14.1: Searching Near Root
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Exercise on the Goal Seek Spreadsheet Feature
initial value x0 , the “starting value”, and computes a sequence x1 , x2 , x3 . . . xn
where xn+1 is determined in the following way:
The function F (x) = 0 is approximated by its tangent3 at the point
(x0 , F (x0 )). The equation of the tangent is
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F (x0 ) + (x − x0 )F 0 (x0 ) = y
Then x1 is the point where the tangent line intersects the x–axis and x1 is
taken as the next approximation to the solution. Thus, for determining xn+1 ,
we get the equation
F (xn ) + (xn+1 − xn )F 0 (xn ) = 0
Solving for xn+1 , we get the equation
F (xn )
F 0 (xn )
8,
xn+1 = xn −
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F (xn ) The algorithm stops when − F 0 (xn ) is less than the largest error one is willing
to permit in the root.
By starting at point x0 , Newton’s method will give us a succession of
values which will eventually converge to the solution. However, consider a
slightly different function, using the starting point shown in Figure 14.2.
In this case, the algorithm may easily get “stuck” to the left of zero
and never find the solution, but rather converge to the location of the local
minimum. It is even possible for the results to diverge, and get farther from
the solution at each iteration!
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14.3.1
Uncertainties in Parameters
One difficulty with using this method for inverting a function is that we
cannot easily generate standard errors in the independent variable. While
there is no general way to do this, we can get some idea of the sensitivity of
the independent variable to changes in function paramter values by varying
the paramter values within their experimental uncertainties. This can be
done as follows, for a function with three parameters A, B, and C:
If we can’t or don’t want to figure out the equation for F 0 (x), we can approximate
it by determining the secant line of F (x) at that point. This is the line which joins
(x, F (x)) and (x + δ, F (x + δ)), where δ is a very small number.
3
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14.3 Theory
F (x0 )
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F (x1 )
6
6
x1
x2
x0
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Figure 14.2: Searching Near Local Minimum
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Exercise on the Goal Seek Spreadsheet Feature
1. Set up new variable values in the spreadsheet so that, for instance,
A0 = A + 2(rand() − 0.5)(∆A)
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(The same can be done for B and C.) (The rand() function of the
spreadsheet gives a random value between 0 and 1, so subtracting 0.5
from it gives a random value between −0.5 and +0.5, and multiplying
this by 2 gives a value between −1 and 1 ; thus we get a value between
A − ∆A and A + ∆A.)
2. Now you will see that any time you recalculate, all of those values (and
thus the results) will change as well.
8,
3. Copy the values A0 , B 0 , and C 0 and use the Paste Special command
to paste the values as numbers, not formulas into somewhere in the
spreadsheet.
4. Change the setup of goalseek to point to the values you just copied
instead of the original A, B, and C.
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5. To see the effect on the parameters calculated, you will need to repeat
the cut-and-paste step and re-run goal seek.
6. Do this 5 or 10 times and use the ranges in the independent variable
values as estimates for the uncertainties in the independent value.
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(You might wonder why we don’t just change all of the values to their values
with uncertainties added, and then see how much that changes the answer,
thus giving us the maximum uncertainty. The problem is that in complicated
calculations, the effects may cancel each other out, so there is no apparent
uncertainty. If we were subtracting the numbers, this is obvious.)
14.3.2
Example: Non-Invertible Functions
Suppose we have a function
y = Ax2 + B/x
and we want to find the value of x for which y = 5. If we want to find the
value of x such that
5 = Ax2 + B/x
14.3 Theory
115
then that is the same as finding the value of x for which
5 − Ax2 + B/x = 0
1. Set up a cell containing the formula
5 − Ax2 + B/x = 0
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In this case, A and B are constants, and we wish to find the value(s) of x for
which this is true, so A and B are our “data”, and x is our “result”.
8,
where A, B, and x are references to variable cells. The first two are
parameters in your equation, and the third is your “starting value” for
x. Provide values for all of these cells.
2. Set up goal seek with x as your variable, and the cell containing 5 −
(Ax2 + B/x) as your formula cell. Make your target value 0.
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3. Run goal seek a few times to see if your results converge. If your results
do not converge, try other starting values.4
To get some idea of the sensitivity of the parameters to changes in data
values by varying the data within its experimental uncertainty, proceed as
follows. (Note in this case the variables are the parameters A and B.)
1. Create new data values above so that, for instance,
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A0 = A + 2(rand() − 0.5)(∆A)
and
B 0 = B + 2(rand() − 0.5)(∆B)
2. Now you will see that any time you recalculate, those two values will
change as well. Copy these A0 and B 0 values and use the Paste Special
command to paste the values as numbers, not formulas into another
part of the spreadsheet.
4
If you know enough about the function to be sure that it has a root, then there should
be some starting value which will work. You just have to find it.
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Exercise on the Goal Seek Spreadsheet Feature
3. Change your formula cell to point to these new copied A0 and B 0 values
instead of the previous A and B. Now the the solution, x will depend
on the uncertainties in A and B.
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If you don’t point to the pasted values, then the random variation will be
happening while goalseek is running, which could make your results very
unpredictable.
4. To see the effect of the uncertainties on the parameters calculated, you
will need to repeat step 2 above and then re-run goal seek.
14.3.3
8,
5. Do this 5 or 10 times and use the range in the solution as an estimate
for the uncertainty.
Goal Seek in Excel
TM
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Goalseek is a tool in Excel, which is simple to use. When you invoke it from
the menu, you will be presented with a dialogue box as shown in Figure 14.3.
f (x) − y0
0
x
Figure 14.3: Goal Seek parameters
Set cell Select the cell for the modified function.
To value 0
By Changing Cell Select the cell for the independent variable.
14.4 Procedure
14.4
Procedure
14.4.1
Preparation
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Pre-Lab Tasks
PT1: Bring a spreadsheet with dispersion data and values for the fit parameters A, B, and C and their uncertainties for Equation 14.1 calculated from
the “Exercise on the Solver Spreadsheet Feature”.
n=A+
14.4.2
B
λ−C
Investigation
(14.1)
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1. Follow the steps from section 14.3.2 using Equation 14.1 and the parameters calculated from the “Exercise on the Solver Spreadsheet Feature”
to find the Helium wavelengths from the values for n which you calculated for them. In this case, you can use starting values for the
wavelengths based on similar coloured lines of other sources which you
know. Put the results in the appropriate columns of Table 14.1.
Inlab Questions
IQ1: What values did you get for the wavelengths? Do they make sense,
given the wavelengths you know?
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Inlab Tasks
IT1: Demonstrate your results to the lab instructor.
14.4.3
Analysis
1. Determine the sensitivity of the wavelengths to the parameters as described in section 14.3.2. Put the results in the appropriate columns of
Table 14.1.
2. Add a series to your graph of n versus λ to show the Helium points,
with error bars.
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Exercise on the Goal Seek Spreadsheet Feature
3. Look up the values for Helium spectral lines and see if the wavelengths
you calculated correspond to observed lines.
Postlab Discussion Questions
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Q1: What are the values you get for the Helium wavelengths, including
uncertainties given by the sensitivity method? Did they agree with accepted
values?
Postlab Tasks
14.5
Recap
8,
T1: Hand in your graph, the calculated results and question answers.
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By the end of this exercise, you should know how to use the goalseek function
to work with non-invertible functions.
14.6
Summary
Item
Pre-lab Questions
In-lab Questions
Post-lab Questions
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In-lab Tasks
Post-lab Tasks
Number
0
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1
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1
Received
weight (%)
0
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14.7 Template
Template
My name:
My student number:
My partner’s name:
My other partner’s name:
Today’s date:
∆n
λ
∆λ
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n
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14.7
119
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Table 14.1: Helium wavelengths from curve
Exercise on the Goal Seek Spreadsheet Feature
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Appendix A
Aligning the Spectrometer
Introduction
8,
A.1
To align the spectrometer you will need the following pieces of equipment:
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• Spectrometer,
• Bubble Level,
• 60◦ Prism,
• Front Surface Mirror,
• small Allen wrench, screwdriver,
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• Variable Intensity White–Light Source.
A.1.1
Levelling the Spectrometer
1. Place the spectrometer on the laboratory table so that it will be convenient to rotate the telescope when making observations.
2. Place the bubble level on the base of the spectrometer and use shims
(at 3 points separated by 120◦ ) to level it.
3. Adjust the height of the prism table so that the prism face is centered
on the telescope and collimator lenses.
Aligning the Spectrometer
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122
8,
Figure A.1: Telescope Adjustment
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4. Place the bubble level on the prism table and rotate the table so that
the adjusting screws are opposite the corresponding shims on the spectrometer base. Level the prism table using the adjusting screws.
5. Rotate the prism table through 60◦ and adjust the shims on the table
so that the prism table is once again level.
6. Rotate the prism table in the same direction through another 60◦ and
adjust the screws so that the table is level.
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7. Repeat the previous two steps until the prism table is level in all positions.
A.1.2
Telescope Adjustment
1. Figure A.1 shows a schematic view of the telescope used in this part
of the experiment. Do not poke anything down the cross–hair tube,
since the cross–hairs may be damaged or moved out of their pre-aligned
position.
2. Make sure that the opening in the eyepiece coincides with the opening
in the cross–hair tube. Adjust the eyepiece until the cross–hairs are in
sharp focus.
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A.1 Introduction
8,
Figure A.2: Collimator Adjustment
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3. Place the front surface mirror across the objective lens and shine light
(from the white–light source) into the opening. Look into the eyepiece
and observe the cross–hairs and their reflected image. Loosen the focusing screw and adjust the cross–hair tube until the reflected image of
the cross–hairs is in sharp focus and there is no parallax with respect to
the real cross-hairs. Lock the tube into this position with the focusing
screw.
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4. Remove the front surface mirror and place the prism on its table with
the polished face perpendicular to the table surface. Shine white light
into the opening and rotate the table until the cross–hairs and their
reflected image can be seen through the eyepiece. Adjust the leveling
screw on the telescope until the cross–hairs coincide with their image.
The telescope has now been aligned to accept parallel light rays.
A.1.3
Collimator Alignment
1. Figure A.2 shows a schematic view of the collimator. Remove the prism
from the table. Open the adjustable slit of the collimator and place
the white–light source in front of it.
2. Rotate the telescope until its cross–hairs are aligned with the adjustable
slit. Adjust the leveling screw of the collimator until the slit is in the
center of the eyepiece’s field of view.
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Aligning the Spectrometer
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3. Loosen the focusing screw of the collimator, and adjust the collimator
tube until the slit is in sharp focus. Now reduce the slit width (do not
close the slit completely!) until the slit image is slightly larger than
the cross–hairs, and remove any parallax between the slit image and
cross–hairs. Lock the collimator into this position with the focusing
screw. The collimator has now been aligned to produce parallel light
rays.