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PHYS 102: TA Lab Manual
Nikolas Provatas, Anh-Khoi Trinh, Cesar Daniel Rodriguez Rosenblueth
November 10, 2018

Contents
I

General Guidelines

1 Lab Format
1.1 Introduction, Learning Objectives and Lab Structure . . . . . . . . . . . . .
1.2 Lab Log Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Data Analysis
2.1 General Guidelines
2.2 Data Presentation .
2.3 Basic Statistics . .
2.4 Error Analysis . . .

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Lab Manuals

3 Lab 1 - Capacitance
3.1 Learning objectives . . . . .
3.2 Introduction . . . . . . . . .
3.2.1 Pre-lab activity . . .
3.2.2 What is a capacitor?
3.3 Protocol . . . . . . . . . . .
3.3.1 Session a . . . . . . .
3.3.2 Session b . . . . . . .
3.4 Measurement Analysis . . .

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4 Lab 2 - RC Circuits
4.1 Learning objectives . . . . . . . . . . . . . .
4.2 Introduction . . . . . . . . . . . . . . . . . .
4.2.1 Pre-lab activity . . . . . . . . . . . .
4.2.2 RC Circuits . . . . . . . . . . . . . .
4.2.3 Equivalent resistance and capacitance
4.2.4 Equipment . . . . . . . . . . . . . . .
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4.3
4.4

Session a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Session b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Lab 3 - Magnetic Field of Solenoids
5.1 Learning objectives . . . . . . . . .
5.2 Introduction . . . . . . . . . . . . .
5.2.1 Pre-lab activity . . . . . . .
5.3 Session a . . . . . . . . . . . . . . .
5.3.1 Solenoid setup . . . . . . . .
5.3.2 Magnetic Field Lines . . . .
5.3.3 Current dependence . . . .
5.3.4 Distance dependence . . . .
5.4 Session b . . . . . . . . . . . . . . .
5.4.1 Electric motor setup . . . .
5.4.2 Protocol . . . . . . . . . . .

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Appendix

A Introduction to Excel

B Lab
B.1
B.2
B.3
B.4
B.5
B.6

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Equipment
Multimeter . . .
Arduino . . . . .
Reading Resistors
Bread Boards . .
IOLab . . . . . .
Oscilloscope . . .

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C More statistics

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Part I
General Guidelines

Chapter 1
Lab Format
1.1

Introduction, Learning Objectives and Lab Structure

Welcome to PHYS 102! The focus of these lab session will be to teach you how to think
critically about data collected from experiments, a skill that can be applied in many other
STEM fields and in your own personal lives. You will learn how to acquire and analyse data
quantitatively in experiments pertaining to electromagnetism experiments.

Learning objectives
Whether it is in your professional or personal lives, the ability to analyse and think
critically about information, particularly information in the form of data, is a valuable one. In
an age where one can feel overwhelmed by the prevalence of information from documentaries,
non-fiction books, online websites and politicians, one must find a way to sift through this
sea of information to determine objective truths. As scientists, we recommend the scientific
method.
While the labs will be related to electromagnetism topics that you will learn about in
class, the main focus of these labs will be for you to be acquainted with the scientific method.
This methodology will be used to answer a question posed to you through data collected and
observations you make. The first step is to conjecture a falsifiable hypothesis to answer this
question. One then performs an experiment to verify said hypothesis. After that, one can
make a conclusion based on the data. In these labs, you will be asked to apply the scientific
method quantitatively to electromagnetism experiments.

Lab structure
In these labs, you will have two 2-hour sessions to complete your labs: one session per week
over the span of two weeks. You will have different tasks to accomplish per session. The goal

of this two-session setup is to allow students to make mistakes during the first session, think
critically about these, and improve your experimental data and or interpretation of these in
the second session. Contrary to other lab courses that you may have taken, the emphasis
here is to evaluate a student’s critical thinking process. This is best achieved when acquiring
crude or bad data at first, and leaving room for follow up and subsequent improvement. The
critical thinking process of improving one’s data is the main focus of these labs.
To evaluate a student’s critical thinking process, we will resort to the following grading
rubric:
• 15% for attendance (to both sessions of a given lab)
• 15% for a pre-lab analysis
• 10% for participation in group discussions
• 60% for the content of your log book
Given the focus on critical thinking, the instructions for each lab will be minimal. Students
must learn how to acquire meaningful data on their own. Their thinking process must therefore
be reflected in their log book with enough clarity, organization and precision to allow us
evaluate their ability to think critically. TAs will be more involved that in typical labs in
order to guide students throughout the labs.
At the start of the first session, each student will be asked to submit their pre-lab answers
to specific questions posed. Students will then be asked to divide into groups of three, and
perform measurements as a team to collect data related to the posed question(s). All methods,
experimental data collected, and interpretations will be written in log book. By the end
of the first session, students are asked to hand in a single log book per team. While you
cannot bring the log book home, you are allowed (and encouraged) to take pictures of the
log book to think about your results between the two sessions and plan ahead what you
intend to do in the second session to improve your experiments. Although you will work
with one teammate, and submit one joint journal, as is common in scientific research, we
highly encourage discussions with other teams and students outside of the lab session. Such
discussions will start in the lab, where TAs will facilitate group discussions between groups
to help students troubleshoot their experiments and understand sources of error. You will
receive your log book again at the start of the second session to document your improved
experiment, new data collected and any revised interpretations. By the end of the second
session, students will again submit their log book for evaluation. For most labs, there will be
an individual work station and a collective work station. The former offers the base materials
and supplies for each group to successfully finish the lab, while the collective work station
offers additional material that can be used to improve your data.

1.2

Lab Log Books

The log book will be your primary source of evaluation. You will be asked to submit a log
book entry by the end of each session. We will distribute at the start of each session the
following structured log book:
[We still have to discuss the grading scheme]
i) Introduction (5%)
Clearly state your goal. This can be a brief re-statement of the question, or of your method
to improve your data. Clearly state your hypotheses. Write down what you intend to verify
experimentally, and why this method is an appropriate verification of your hypothesis. This
part may differ for the second session since you may be more acquainted with the theory and
experiment.
ii) General Notes: Methodology and Rough data (35%)
In this section, we expect you to write all your observations so that we can see your thought
process as you are performing the lab.
As opposed to other lab reports that you may have submitted, we do not expect a clean
point-by-point list of manipulations. In fact, you may start following a set of protocol and
realize you made some mistakes and therefore should change your original protocol. We
recommend writing short sentences and/or intelligible and readable notes of what your are
doing. If you do encounter unexpected problems, please write down what they were, and how
you resolved them.
We encourage you to write your measurements and rough calculations as you do your
measurements. If you choose to compile your data in Excel, you should simply write a
few sample calculations. These sample calculations do not need to follow the significant
figure rules to be outlined later. More importantly, you should write down all your
observations. This part can be messy, but as long as the information that you want to
convey is clear. We recommend boxing, colouring, underlining and/or labelling important
measurements, calculations or observations.
iii) Results (30%)
In this section, you will present your data in its most presentable form. This differs from
the previous section by the way that you present your data: significant figures, graph rules,
etc. have to be followed. You will also state your observation and analysis in this part. We
expect you to be able to present your data adequately according to instructions from the
Data Analysis section. For electronic files such as Excel documents, a TA will have a USB
drive onto which you are to upload your documents.
iv) Conclusions and future work (30%)
6

In this section, you will state your conclusion, and offer possible follow-ups to this lab. This
can include other physical properties to study, or to offer a better methodology to study this
phenomenon. You must motivate your proposed follow-up project by clearly stating what you
expect to observe (a hypothesis) and support your claim based on your current experiment.
The proposed future projects must be realistically performed by your peers, and must not
assume an unrealistic sum of budgetary support or higher level of experimental expertise
such that you or your peers are not acquainted to. For your lab report submitted at the end
of the first session, be mindful that you must execute your proposed improvements during
the second session.

Chapter 2
Data Analysis
2.1

General Guidelines

Here is a list of general guidelines for your log book:
• Measurements must always include units.
• Measurements must include an uncertainty (standard deviation).
• Tables (and any other data representation method) must also include appropriate labels,
units and standard deviation.
• When comparing to a known quantity (theoretical value), the former does not have a
standard deviation attributed to it, unless that information is found from an experiment,
in which case there should be an associated standard deviation.
• If you quote results from elsewhere, you must include the source: [how to quote sources]
• Recall that vector-data has both a magnitude and a direction.
• Write down all your observations and some commentary about them.
Figures must
• Include a title.
• Include labels on your axes.
• Include units on your axes.
• Include a few tick markers for the audience to know the scale.
• Use an appropriate scale.
8

• Include a legend for the different curves shown on a single plot.
• Fit equation (and R2 value) if a fit was used.
• Not have joint data points (unless necessary).

2.2

Data Presentation

One of the most important aspects of doing research is to be able to communicate efficiently
your findings. In physics, this is done mostly by means of one of the following:
• A quoted result
• A table of results
• A figure
As emphasized previously, all quoted results must include an uncertainty i.e. a standard
deviation and units. For example, if I measured the gravitational acceleration, I can quote it
as
g = 9.8 ± 0.2 m/s2 .
(2.1)
Usually, if you have a single result, you must explain what it is: was it from a single
measurement? Was it an average or did you obtain it from some other method? Whatever
the method that you used, there is a specific way to obtain the correct uncertainties. This
will be explained in the next sections.
On rare occasions, there are certain quantities that do not need a standard deviation.
This could include percentages of error or absolute error differences.
If you choose to present multiple results, you may opt for a table. In this case, you still
must include standard deviations and units.
Voltage (±0.2 V)
3.1
5.4
7.3

Current (±0.1 A)
15.5
25.2
35.6

Table 2.1: Example of a table where all measurements have the same standard deviation. If
they didn’t, you must still include them individually.
Tables are useful to show the explicit values of your results. However sometimes the exact
values aren’t as important as the general trend. In this case, you would opt to present your
data in chart form.
There are many types of possible charts. The most useful one for us will be the scatter
plot. An example of a scatter plot is shown in Fig. 2.1.
9

Figure 2.1: Example of a scatter plot. Each measurement is displayed as a point. One can
add error bars in both the x and y axes (although only the y direction is displayed here).
One can also fit an equation to the data set.
A scatter plot allows you to see the general trend of your data. If you expect a specific
behaviour, you can fit a trendline to your data to match a specific equation. By further
plotting error bars, you can compare how far away your data points are from the expected
trendline. This gives you some insight into whether your data matches your expected trend.
From the trendline, you can also see the actual coefficient of the fit. As shown in Fig. 2.1,
you can also display a R2 value, which is called the coefficient of determination. This tells
you how good your fit was to your data. A perfect R2 value is 1. For the example of Fig. 2.1,
the R2 value is good and most of the measurements fall within a standard deviation of the
fit, therefore this says that the current is proportional to the voltage by a proportionality
factor of 3.0371.
In summary, to present your data, you will need to know certain things: how to perform
an average and other basic statistics, how to evaluate the uncertainty in your measurements
and by extension, your final result, how many digits to display in your final answer, and most
importantly, how to analyse your data. This will be the subject of the following sections.

2.3

Basic Statistics

Mean
Let y(x) be a function of xi datapoints. The value of y for xi can be denoted by y(xi ) or

yi . The mean, or average, denoted by ȳ of a dataset with N datapoints is given by
N
1
1 X
y(xi ) =
(y(x1 ) + y(x2 ) + ... + y(xN )) .
ȳ =
N i
N

(2.2)

Standard deviation
The standard deviation of a function y(x) is denoted by σy . One says that a measurement
y(xi ) has an uncertainty of
yi ± σyi .
(2.3)
The standard deviation tells you up to what value is your data precise: your data point can
vary between yi + σyi and yi − σyi . This means that your measurement yi can vary by 2σyi .
A good set of measurement would have a very small standard deviation: this gives you the
bounds under which your measurement is considered precise. You will see in section “Error
Analysis” how to quantify this quantity.

Significant Numbers
To present results, one must present an appropriate number of significant numbers. In
this course, we will not be concerned with the number of significant digits, but more with the
precision attributed to a measurement. When quoting a measurement, the last digit should
be fixed by the uncertainty of the measurement: it must match the order of magnitude of
the standard deviation. In other words, the standard deviation fixes up to what precision
(roughly decimal place) you can quote your results. Table 2.2 lists some common mistakes
and how to present those results appropriately. In the context of this course, we will simplify
our conventions by keeping only one significant number for the standard deviation. Recall
that leading zeros do not count as a significant number and trailing zeros (after the precision
of the standard deviation) should be discarded.
Incorrect
324.9453 ± 0.02m
20 ± 0.1$
3 × 104 ± 2N
31.02 ± 0.10$
0.003564 ± 0.00012m/s

Correct
324.95 ± 0.02m
20.0 ± 0.1$
3000 ± 2N
31.0 ± 0.1$
0.0036 ± 0.0001m/s

Table 2.2: Significant numbers. Common mistakes and the correct way to present such
results.

Precision versus Accuracy
Precision refers to how closely related two measurements are from each other. This
means that one expects a small standard deviation for highly precise measurements.

Accuracy measurements refers to the closeness of a measurement from a known value.
This means that one must calculate the difference of the measurement to the known value
in terms of standard deviations. For example, if y = 0.4 ± 0.1m, and the known value is
yt = 0.7m, then the measurement differs by 3σy .
Often, one may encounter a highly precise set of measurements, but also very inaccurate.
This is a signal that there may have been a systematic error in your results. This is a type
of error that propagates throughout your experiment and offsets your results by a constant
factor. For example, recall the equation of the period of oscillation of a pendulum is
s
T = 2π

L
.
g

(2.4)

If one incorrectly measured the length L of the string, then the results would be offset by a
p
constant factor of Lgood − Lbad .

2.4

Error Analysis

Standard deviation of a measurement
The rule of thumb for finding the standard deviation of measurements is as follows:
each measurement can vary by half of the smallest available precision from your measuring
instrument, and you must add the uncertainties of the measurement. Consider the example
of measuring a distance with a 30cm ruler. The smallest increment is a mm. When you
measure a distance between two points, you take two measurements: one at each point. Thus
the uncertainty on the measurement is 2 × 0.5mm, so 1mm. This is exactly the smallest
available increment of your measuring device. Therefore, unless otherwise specified, the
standard deviation of a measurement is the smallest available measurable value allowed by
your instrument.
Another example is measuring the time difference between two events. Again, there will
be two measurements: the initial and end time. Stopwatches often allow one to view up
to ms precision. However, the average visual time reaction is roughly 0.3s, and therefore
each initial and final time measurement varies by 0.15s and so the uncertainty for the whole
measurement is 0.3s.
Standard deviation for fluctuating data
For fluctuating datasets where each measurement is independent of the previous, to reliably
represent your data, you should take some form of average over some number of measurements.
Measurements that are random are said to follow a Poisson distribution. To study these
types of measurements, one typically chooses a subset of N data points, and calculate its
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mean. Given the mean, one must approximate the error as
di = yi − ȳ.

(2.5)

The standard deviation on the average of your measurements will therefore be given as
v
u
N
u1 X
t
σȳ =
d2i .
N i=1

(2.6)

This can be obtained in Excel by using the function =STDEV.P(x).

Error propagation
The standard deviation value above is attributed to a single measurement. However, often
one would be interested in deriving a quantity that depends on multiple measurements. To
describe the standard deviation on the end result, one must use error propagation techniques.
For these labs, you will only need to know the following two error propagation formulas.
Consider first the following function
f (x, y) = x × y.

(2.7)

The standard deviation σf on f will be
σf =

q
(xσy )2 + (yσx )2 .

(2.8)

This will be useful for example to calculate the error on the area given two length measurements,
or in Lab 2, you can calculate the error on the time constant in a similar way.
Consider also the function
1
f (x) = .
(2.9)
x
The standard deviation for this case is
σf =

σx
.
x2

(2.10)

Measurement analysis
To understand the significance of a measurement, one can compare it in different ways.
When a theoretical value is known, one can compare a set of measurements by calculating
the absolute error, the percentage error or the standard deviation error. One can also analyse
a dataset by studying fits in a similar manner.
Absolute error

The absolute error indicates the difference between the known value, and either a measurement, the mean value of a set of measurements, or a fit parameter. One can also compare
a set of measurement to the fit of the dataset in this matter. Let yt be the theoretical value
and y be one of the variables mentioned before, then the absolute value is
Absolute error = |yt − y|

(2.11)

Percentage error
The percentage error is simply the ratio of absolute error relative to the known value.
This gives an indication, in percentage, of how far away with respect to the known value is
the measurement. It is given as
Percentage error =

yt − y
× 100
yt

(2.12)

Standard deviation error
The standard deviation error indicates how many standard deviations away is a measurement when compared to the known value. It is given as
Standard deviation error =

yt − y
σy

(2.13)

You will have to recognize which one of the above is useful and representative of the
information that you wish to convey for the data analysis that you have to perform. Note
that we’ve taken the absolute values of the above, but by not doing so, we could convey
information about the “direction” of the error i.e. whether the measurements are higher or
lower than the known theoretical value.

Fit analysis
As mentioned before, a good way to know whether your fit is correct is to compute the
coefficient of determination R2 . To compare your fit to your data, you should see whether
your data points all (or mostly) fall within one standard deviation of the fitted equation, that
is if the standard deviation error should be less than 1 σ for all data points when compared
to yf it .

Part II
Lab Manuals

Chapter 3
Lab 1 - Capacitance
3.1

Learning objectives

• How to acquire data
• How to quantify error
• How to present data
• Learn about linear fits
• Confirm capacitance law

3.2

Introduction

In this lab you will be asked to accomplish the following tasks:
• Find possible parameters that could affect the capacitance of a parallel plate capacitor
system.
• Support your conclusion based on a quantitative evaluation of your data.
You will have to chose some parameters and quantify their affect on the capacitance of
the setup. In the process, you will learn how to acquire good data and how to quantify and
present your data. We would like to re-emphasize that collaborations and discussions are
encouraged and expected throughout the two sessions.

Role as a TA
Your role as a TA will be to guide them through the “script” that we present in
the protocol section. You should guide them by asking questions and making
them think about their results as opposed to giving them the right answer
immediately. You will also be required to bring teams together to collaborate
with each other. You should clarify the grading method, and re-emphasize
that the major part of their grade comes from the log book.
In the first session, you will work together to find possible parameters that can affect the
capacitance of a pair of parallel plates. Measurements at this stage may be crude; do not
worry about that, we are not evaluating the usefulness of the data at this stage, rather the
process of thinking about what this data means, and your thinking process about how to
improve your experiment to obtain better data.
During the second session, you will be asked to obtain quantitatively improved data
and to compare it with data from the previous session. You must conclude, based on a
quantitative evaluation of your data, whether or not, and how, your chosen parameter affects
the capacitance of a parallel plate capacitor.

3.2.1

Pre-lab activity

To answer some of these questions, we encourage you to read the Data Analysis chapter of
the lab manual.
1. What is a measurement? Give your own definition of what constitutes a measurement
based on your previous courses with lab components (biology, chemistry, physics).
Answer
This is more open-ended. Measurements can be taught of as quantifying
observations. Measurements must come with an inherent uncertainty.

2. Two engineers must measure the distance between the McGill metro station to the
athletic centre to build an underground tunnel. One engineer finds that the distance is
9134 ± 5m while the other finds that the distance is (915 ± 2) × 101 m. In what ways
can they compare their measurements? Do their measurements agree? Explain.
Answer
They should compare the standard deviations to conclude that the measurements of the two engineers agree.

3. The city considers to build a metro line between McGill station and the station at
University of Montreal. To do this, an engineer decides to use a km-long ruler whose
smallest divider is a meter. She did not use any other measuring devices. She takes
her measurements and presents her findings to her team as 3653.4500m. What is the
uncertainty on this measurement? Did she present enough, too little or too many digits
about her measurement? Why?
Answer
The correct way to present this measurement is 3653 ± 1m since the smallest
increment on her measuring instrument is a meter. She presented too many
digits since her measuring tool cannot give such precision.

4. Two groups of researchers are trying to test their new methodology to measure the
speed of light c. Current data shows that the speed of light is roughly c = 2.99792 × 108
m/s. The first group finds that c = 2.88 ± 0.02 × 108 m/s while the second group
finds that c = 3.0 ± 0.2 × 108 m/s. What do these measurements tell you about their
methodology? What type of mistake may have caused the discrepancy in the first
group’s results?
Answer
These measurements show that the first group has a more precise methodology, while the second team has a more accurate one. The first group
probably has a systematic error that causes this offset.

5. How do you know if an experimental result is acceptable and trustworthy? What gives
you confidence that your data is trustworthy?
Answer
They should say something about the standard deviation and fit coefficient of
determination. Ideally, they would mention that the methodology must be
questioned as well.

3.2.2

What is a capacitor?

You have learned in class that charges can attract and repel each other. This is because point
charges source electric fields that apply a force on a test charge. Since there can exist a force
between two point charges, there exists a potential energy between two point charges ∆U due
to this force. We define the electrical potential as the potential energy change per unit charge
18

as
∆V =

∆U
q

(3.1)

where q is the test particle’s charge. In practice, it is more useful to define an electrical
potential difference to move any test charge q from r = ∞ to a distance R from a source
charge Q as
kQ
(3.2)
V =
R
where k is Coulomb’s constant k = 1/4π0 . This will have units of voltage V. This process is
depicted in Fig. 3.1.

Figure 3.1: The electrical potential due to a source charge Q (left) at a distance R. The
test charge (right) is placed at a distance r from the location that we measure the electrical
potential difference.
Now consider a pair of parallel plates separated by a distance d connected to a battery
as shown in Fig. 3.2. A pair of parallel plates separated by air or another material can
be assembled into a capacitor : on each parallel plate, because of the battery, there will
accumulate a number of electrons on the negative plate such that the total charge is −Q and
+Q on the positive plate. Because of the gap between the plates, there now exists an electric
potential between the plates, which is given by eq. (3.1).
One defines the capacitance of a capacitor as the constant of proportionality between the
total number of charges Q and the electric potential V stored between the plates, i.e.
Q = CV.

(3.3)

Capacitance is measured in Farads F. A useful constant in such setup is the vacuum permittivity
= 8.85 × 10−12 F/m. In this lab, you will be asked to verify what affects the capacitance C
of a parallel plate capacitor system.

Figure 3.2: Parallel plate setup. The battery with voltage V creates a flow of electrons such
that there will accumulate a total charge of −Q on one plate, and +Q on the positive plate.
Since the plates are separated by a distance d, there will be a electrical potential difference
between the two plates.
Expected result
We expect them to find
A
C= ,
d
where is the relative permittivity. They should be able to test the area
correlation with the plates given, and the distance one by stacking multiple
sheets of dielectric between the plates.
We expect them to plot their data and fit a linear regression with standard
deviation error bars as shown in Fig. 3.3. Knowing the distance between
the plates, they can also plot the relative permittivity versus the area, and
compare with the mean value of to see the consistency of their results as
shown in Fig. 3.4.

Figure 3.3: [Figure only available to TAs] Data of capacitance versus area. We expect them
to display the equation and R2 value. They should be able to find a good linear fit.

When testing for the d dependence, the data showed that for n > 3 sheets,
√
there seems to be a 1/ d correction that appears, and therefore students
aren’t expected to find a perfect fit for large distances of d. This is mostly
due to some polarization effect from stacking the dielectrics, and therefore
one should view the 1/d dependence as a “small distance” approximation.
You should encourage students to find this disparity between experiment and
theory and encourage them to discuss the cause of this. The data is shown in
Fig. 3.5.
Students are encouraged to discuss about other parameters such as shape,
volume, mass applied to the dielectric, and the use of other dielectrics, but
these aren’t easily quantifiable parameters.

Figure 3.4: [Figure only available to TAs] Data of the relative permittivity versus the area,
compared to the mean value. The average is 3.4 ± 0.3 × 10−12 F/m.

3.3

Protocol

3.3.1

Session a
Session #1
We suggest the following time distribution for this session:
During the first 20mins, assemble everyone to the front and do a quick
introduction, and maybe a demo of the manipulations. You should then ask
everyone what they think could affect the capacitance of a capacitor, and
push them to give a physical reason behind their hypotheses. We expect you
to motivate the idea that electrical potential is given by the relative distance
between two charged objects, and therefore, for parallel conducting plates,
larger area would allow for more charges to accumulate, and by eq. (3.2), we
expect there to be a 1/r dependence. Recommend that they works together
to cover more parameters e.g. one team takes care of area, the other does
distance.

Figure 3.5: [Figure only available to TAs] Data of C/A versus 1/r. The data points in orange
are those of the first three layers. The linear fit matches the average found from the area
experiment.

Give them about 60-75mins to perform their measurements while walking
around, answering questions and facilitating discussions between teams and
teammates. During this period, make sure that everyone makes at least three
measurements of varying area with the two plates perfectly aligned. This will
serve as a baseline to put them on the right track. Also, do not forget to
verify that students have done their pre-labs.
After about 90mins of the 2-hour session, call everyone to the front and
ask people to describe their findings. List on the board what could affect
capacitance and why. Make sure that they support their findings with statistics
(standard deviations and fits), and ask them whether the correlations that
they find are well supported and warrant extra verification during the next
session. After this discussion period, make sure that everyone understands
that they have to be looking to verify the effects of area and distance of the
plates during the next session.

For the remainder of the time, students can try and take additional measurements, and prepare notes for the next session. Don’t forget to pick up every
teams’ log book by the end of the session.
The available material is listed in Table 3.1. The metallic rectangular plates will be
of varying dimensions. The manipulations are straightforward: insert either a plastic or
paper sheet between the two plates and use the Arduino unit (see below) to measure the
capacitance. At this point, you do not need to know how the Arduino unit works, simply
how to use it. You will learn how it works in the next lab. The plates must not touch each
other in order for you to see a measurement: this causes a short-circuit and therefore you will
not see a capacitance measurement. You can assume that the distance between the plates is
the thickness of the plastic or paper sheet.
Individual station
Laptop
Arduino capacitance setup
Metallic plates
Ruler (?)
Plastic sheets
Paper sheets

Collective station
Aluminum foil
Blocks of wood
Small weights
Assorted capacitors
Assorted resistors
Extra wires
Micrometer
Multimeter

Table 3.1: Available material for Lab # 1
Information about the Arduino code can be found here. To use the Arduino unit:
1. Plug the USB-port into the laptop
2. Open the Arduino Genuino application on the laptop.
3. Open the capacitanceMeter.ino file found in [location].
4. In the tabs above, go into the drop-down menu Tools → Port. Choose whichever COM
is labelled by “(Arduino/Genuino Uno)”.

Figure 3.6: Arduino application menu bar. Note that the color boxes were added to guide
the eye. The red checkmark box is the verify button. It serves to verify whether there is an
error in the code. The orange right arrow box is the upload button, and the purple box is
the Tools drop-down menu bar.
5. Verify the script by clicking the icon with a checkmark. If there is an error, call the TA.
6. Upload the script by clicking the icon with the right arrow.
7. In the tabs above, go into Tools → Serial Monitor (or on your keyboard, press
Ctrl+Shift+M).

Figure 3.7: Arduino Serial Monitor. This is a pop-up screen where you will see your
capacitance measurements scroll by. The Arduino unit takes a capcitance measurement at a
fixed frequency. Unless otherwise specified, the output is in nanoFarads. Units aren’t display
as to simplify the transfer from the monitor screen to Excel, but one could display the units
by changing the command in the Arduino script. To stop the scrolling of measurements, click
on the Autoscroll box.

8. Take the two wires (see Fig. [updated setup]) and simply touch each side of the parallel
plates: you should see a multitude of measurements scrolling down the monitor window.
If you do not see many measurements scrolling in the monitor window, it is likely that
the plates are touching each other and you therefore have a short circuit. [input new
setup pic]
9. You can stop the auto-scroll by clicking the bottom-left box to better see your results
(see Fig. 3.7).
Notice that in Fig. 3.7, the data fluctuates. You will have to determine how many
capacitance measurements should you average over to acquire a good representation of that
capacitance measurement. We refer you to the Part I, section 2.4 of the lab manual. Using
this setup, proceed to conduct you experiments and collect data that will let you evaluate C
in eq. (3.3).
You will have the opportunity to test whatever hypothesis you may have, however every
group must take a capacitance measurement for at least three different pairs of
plate sizes while inserting a single sheet of plastic between them. The plates
must be perfectly aligned on top of each other. After doing so, you may explore
in more details the consequences of these measurements, or you may study other possible
parameters that could affect the capacitance.
The three area measurements will serve as a baseline for students in case some
of them attempt to test unconventional hypotheses.
Given the measurement of the two sides L1 and L2 of a rectangle and their corresponding
standard deviations σL1 and σL2 , the standard deviation on the area is given by
q
σA = (L1 σL2 )2 + (L2 σL1 )2 .

(3.4)

While studying other parameters, if you are unsure of the standard deviation of a certain
parameter, you may ask the TAs to help derive the appropriate derived standard deviation
for that parameter.
To analyse your data, we refer you to section 3.4. Enter your set up, measurement process
and data in your log book. Before handing in your log book, be sure to write down your
observations in the General Notes section of your log book, and a rough plan of what you
intend to do during the next session in the Conclusions and future work section. The
latter part should be done after discussions with at least one other group.

3.3.2

Session b

We recommend that you bring whatever material that you think could help you acquire better
data. At this point, you should know what parameters affect the capacitance, and therefore
you should plan in advance your own “protocol” to verify your hypothesis quantitatively
with improved data. This protocol does not have to match what you wrote in your log book
during the last session, but if it does differ, you must explain what made you change your
mind.
Session #2
In this session, you should remind everyone that last time, you discovered
together that C ∝ A/d, so this session should be simpler as they will know
what to do.
Be mindful that you will be evaluated based on your data presentation, data analysis and
explanation of the physics, therefore be certain to fully, but succinctly, describe your thought
process.

3.4

Measurement Analysis

During this lab, you will have to use data analysis techniques to understand your data. For
each session, we recommend that you analyse your data as follows.
To study a system, one typically isolates as many variables as possible to see each of its
individual affect on the whole system. If we are able to isolate such a variable, mathematically,
that is the equivalent of finding some f (x) that depends just on one variable x. In this lab,
you will find as many variables that can affect the capacitance as possible by isolating each
parameter one at a time.
For each dataset, corresponding to each parameter you measure, you must abide by the
error analysis rules outlined in chapter 2. The data analysis can be done much more rapidly
with spreadsheets, therefore we highly recommend that you read appendix A.
Once you have some result from your data, you must offer a physical explanation that
supports your results. Explain why physically each parameter affects the capacitance of the
system. Support your claim by proposing a possible experiment to test your claims, and offer
possible outcomes that one could observe from your experiment. Recall that bouncing your
ideas off peers is an invaluable way to actually test the rational of your hypothesis.

Chapter 4
Lab 2 - RC Circuits
4.1

Learning objectives

• Learn how to connect circuits
• Learn about exponential functions
• Learn about RC circuits
• Learn about equivalent resistance and capacitance

4.2

Introduction

In this lab you will characterize an RC circuit. An RC circuit is a circuit that contains
a resistor (R) and a capacitor (C). The analysis tool we will use is called a PASCO 850
Universal Interface, the use of which will be explained below. Specifically, we will also use a
voltmeter and a signal generator tool to examine the charging and discharging of a capacitor.
In the first session, you will be asked to measure the time constant τ of a given
RC circuit. In the process, you will learn how to measure time constants and understand
subtleties in the data acquisition apparatus. This session will be a more traditional lab where
you will follow well outlined instructions.
In the second session, you will be asked to build an RC circuit with a specific time
constant and to show, based on well supported data, that your circuit has the appropriate
time constant. When you arrive at the second session, you will receive a sheet that asks you
to build an RC circuit with a specific number of parallel branches. You are to choose an
appropriate number of resistors and capacitors to obtain a specific time constant.

4.2.1

Pre-lab activity

1. Why would we want to fit a dataset to an equation?
Answer
Curve fitting allows one to match data to a quantitative theoretical model. A
good model is able to predict future outcomes of similar experiments.
2. What is ex evaluated at x = 0, x = 1 and x = −1?
3. For f (x) = e−x , what is the value of x so that f (x) = 0?
4. What does a, b and c do in the following function?
f (x) = aebx + c

(4.1)

5. What is the difference between these two plots:

Figure 4.1: Exponential functions.

4.2.2

RC Circuits

Recall that the capacitance is defined as the proportionality constant between the total charge
accumulated by a capacitor and the voltage across the circuit
Q = C∆V.
29

(4.2)

In this equation, the charge Q is expressed in Coulomb (C), the voltage ∆V , in volts (V)
and the capacitance C in farads (F).
In this lab you will study both the charging and discharging process of an RC circuit.
During the charging process, charges accumulate on each sides of the parallel plate capacitor.
During the discharging process, the capacitor releases all its charges into the circuit. Capacitors charge and discharge exponentially in time. During the discharge of a capacitor, the
instantaneous voltage ∆VC between the ends of the capacitor also drops and is given by
∆VC = ∆Vmax e−t/τ

(4.3)

where ∆Vmax is the maximum voltage across the capacitor, i.e. the voltage to which the
capacitor was charged, t is the time and τ is the time constant given by
τ = Req Ceq

(4.4)

where Req and Ceq are respectively the equivalent resistance and capacitance of the circuit.
Although the theoretical discharge time is infinite, in practice we consider that the discharge
is over when the voltage at the bounds of the capacitor is at 1% of its maximal value.

4.2.3

Equivalent resistance and capacitance

To find the equivalent resistance and capacitance of a circuit, one must apply the correct
equations to sum the contributions of all the components. The equivalent quantity differs
depending on whether the resistors and capacitors are combined in series or in parallel. For
resistors, the equivalent resistance is given as
Rseries = R1 + R2 + R3 + ...Rn =

n
X

(4.5)

i=1

1
Rparallel

X 1
1
1
1
1
=
+
+
+ ... +
=
R1 R2 R3
Rn
Ri
i=1

(4.6)

For capacitors, the equivalent capacitance is given as
Cparallel = C1 + C2 + C3 + ...Cn =

n
X

(4.7)

i=1

1
Cseries

X 1
1
1
1
1
=
+
+
+ ... +
=
C1 C2 C3
Cn
Ci
i=1

(4.8)

4.2.4

Equipment

In this lab you will use the PASCO 850 Universal Interface to measure and plot voltages as a
function of time in an RC circuit. The PASCO devices comes with two sets of wires (see
Fig. 4.2). The first is the input/output voltage source. The other is a set of probe which
measures the voltage across the capacitor of the circuit.

Figure 4.2: PASCO 850 Universal Interface The red box is the probe Analog input. The
voltage source (Output 1) is boxed in orange.
Given the nature on these measurements, there is an inherent fluctuation to the data
that is difficult to characterize. During the first session, you are not required to produce
quantitatively well supported data. But for the second session, you are to demonstrate that
you’ve produced an RC circuit with the correct time constant.

4.3

Session a

Equipment
During the first session, you will receive an RC circuit fastened onto a wooded board. Before
proceeding with measurements related to charging and discharging capacitors, conduct the
following steps:
1. Turn the wooden board over to reveal the circuit Fig. 4.3. There are actually two
circuits here, one consisting of a 100kΩ rheostat (variable resistor) and the 100µF
capacitor, the other consisting of a 470Ω resistor and a 0.1µF capacitor. We will not
use the 0.1µF capacitor, or the rheostat.
2. Connect the Output 1 of the PASCO 850 Universal Interface to the wooden board’s
terminals as well as the Analog Input A in order to complete the circuit as shown on
Fig. 4.3. The 100µF capacitor can now charge through the 470Ω resistor (when in
the “charge” state) and discharge through the 150kΩ resistor (when in the “discharge”
31

state). The charging voltage is read on the computer. The Input A measures the
voltage through the 100µF capacitor. The Output 1 is the one with the red and black
connectors and the analog inputs are the higher half of the interface and on the right.

Figure 4.3: Circuit for session 1. This is currently connected to the rheostat.
Session #1
In this session you will supervise students as they perform the protocol below.
You may gather students in the beginning to give a brief introduction and
summary of the lab. By the end, they should be comfortable measuring
charging/discharging time constants. When they use the square wave input,
make sure that they understand how to vary the parameters to obtain useful
data, i.e. changing the frequency of the function generator and the frequency
of data acquisition.

Procedure
You will execute the following steps in order to understand how to use the PASCO interface to
characterize an RC circuit. For this session, we recommend that you record your measurements
in an Excel spreadsheet, and rewrite the relevant results in your log book. You are expected
to write down your observations in your log book and to answer the questions that appear
in the following instructions. You should write down all that technical notes about the
apparatus and how an RC circuit works in order for you to fast-track your manipulations for
the next session.
1. Connect the circuit as shown in Fig. 4.3.
2. [Where to get Pasco file]
3. Turn on the PASCO and plug in the USB into the computer.

4. In the software, open the “Signal Generator” tool, which is located on the left hand
side panel and make sure that the signal is set to ”Positive Square Wave” with an
amplitude of 6V and frequency of 0.5 Hz. This is equivalent turning on and off a DC
voltage 2 times per second. Change the data acquisition frequency found at the bottom
of the interface to 1kHz. See Fig. 4.4.

Figure 4.4: Settings for a positive square wave input. You must first select the “Signal
Generator” setting in the yellow box. The red box allows to change the waveform. Change
the frequency and amplitude according to the orange box. Change the data acquisition
frequency to the values in the green box. Once you are ready to record your data, select the
record button (blue box), and turn on the voltage source (pink box)

5. With this input from the ”Signal Generator” you won’t use the ”charge” and ”discharge”
switch from the wooded board. Instead you will simply turn On and Off (pink box) the
power supply under the 850 Output 1 panel before and after recording measurements.
So leave the switch on “charge” for the rest of the experiment.

6. After turning on the signal generator, press the record button twice to record and stop
the recording after observing a few charges/discharges.
7. Zoom in on a region with both charging and discharging behaviours by rescaling the
axes, and figure out which section corresponds to charging/discharging.
8. For the charging part, measure and record the time difference between the moment
the voltage starts increasing to the moment where the voltage is at a value 1 − 1/e of
the maximum voltage V0 . You should understand why you are measuring this time
difference.
9. For the discharging part, you will use the Curve fitting tool. The software can perform
an exponential regression on the data you record. To make good use of that, activate
the Curve Fitting option (see Fig. 4.5). The curve of eq. (4.3) will be fit to the whole
data set. To fit it on the right segment on the data set, use the highlighting tool and
select/box the values that are most relevant. Verify that the time constant corresponds
to the parameter 1/B in the curve fitting data window.

Figure 4.5: Graphing toolbar. The toolbar appears when moving the mouse to the top of
the graph. The red boxed icon is the Curve fitting tool. Select the “Natural Exponential”
function. The highlighting tool is boxed in green. This allows to select a smaller subset of
data to apply the curve fitting.
Use this to fit an exponential curve to the discharging region, and record the value of
the parameter B. You will obtain the discharge time constant.
10. Compare the time constants obtained from both the charging and discharging regions
and by keeping track of the standard deviations as mentioned in section 4.2.4. Calculate
the theoretical time constant based on the known values of resistance and capacitance.
Recall that the standard deviation of the product of two variables is given by eq. (2.8).
Can you explain difference in standard deviation, and why some are smaller/larger than
the others? You can repeat steps 6-10 for different segments of charging/discharging
in your data if you want to verify the consistency of your measurements.
11. Answer the following questions in the results section of your log book. Should all the
time constants be the same or different? Why and why not? What would happen
if we had used the rheostat instead of the 470Ω resistor in Fig. 4.3 and used a DC
voltage source (with the On/Off switch on the board) instead of a wavefunction source:
would we have the same time constant for both charging and discharging? What would
34

happen to the time constant if we increased the resistance by using the rheostat? If
you have time, you should test your hypotheses experimentally.
Answer
All the time constants should be the same because we are charging and
discharging across the same resistor. If we had used the rheostat with a DC
voltage and the On/Off switch, since we’re charging and discharging across
different resistors, the time constants should be different. By eq. (4.4), the
resistance should affect the time constant linearly.

12. Answer the following in the results section of your log book. Assuming the voltage,
when completely charged, is set to V0 = 1 and by considering the variables τ for time
constant and t for time, what are the equations for of charging and discharging? Support
your answer by physical arguments.
Answer
Discharging is V = e−t/τ and charging goes as V = 1 − e−t/τ . At t = 0, we
expect the charging function to be 0 and the discharging function to be 1.
Similarly, at t → ∞, the reverse must be true.

13. Answer the following in the results section of your log book. Your draft calculations
can be shown in the “General Notes” section of the log book. In the circuit of Fig. 4.6,
how will the fully charged capacitor affect the current and potential in R6? Answer
this in your log book.
Answer
The current is defined as the change in charge over time. In other words,
dQ
dV
=C
= I.
dt
dt

(4.9)

Therefore if the capacitor is fully charged, the change in voltage vanishes and
so the current also vanishes: it is then considered like an open-circuit. Thus
the potential across R6 vanishes.

14. Answer the following in the results section of your log book. Your draft calculations
can be shown in the “General Notes” section of the log book. Given Fig. 4.7, what
must be the value of C2 such that the time constant is approximately τ = 12.5ms? If
one replaces C2 by two capacitors in series, one of which has C5 = 250µF. What must
be the capacitance of the second capacitor to preserve the same time constant?
35

Figure 4.6: Effect of a fully charged capacitor on the circuit.
Answer
For the first part, C2 ≈ 50µF. For the second part, C2 ≈ 61µF.

15. If you want to keep a trace of a graph produced by the PASCO, click on the Final
results tab and select which graph you want by choosing the right run in the scrolling
menu of the colored triangle. Click on the camera icon on the top panel. This will take
a picture that will be accessible in the journal, on the right hand side, where it can be
saved as jpeg or html. Alternatively, you can also save your whole set of measurements
by going to the top left tab “File”, and select “Save As”.
Once you have completed the above manipulations, make sure to note important technicalities of the time constant measuring methodology in order to accelerate your data acquisition
process for the next session. You should perform any further tests that you believe would
help you better understand the RC circuit in preparation for the next session.
You may want to pull their attention towards quantifying their errors, in
preparation to the next session. Note that the standard deviation from the
PASCO fit is significantly smaller than that obtained from measuring the
time difference at V = V0 (1 − e−1 ). Discuss the disparity between the error of
each methods.
Resistors and capacitors can be found in the lab. Instructions about reading the resistance
color-code for resistors can be found in the Appendix. We recommend that you try and
read the resistance of a few resistors and confirm by using the in-lab multimeters. An
36

Figure 4.7: Time constant practice.
exemplary circuit board that you must use during the next session will be available for you to
contemplate. We recommend that you observe the set up of the circuit board and think about
how you would connect the resistors and capacitors to obtain different equivalent resistance
and capacitance setup.

4.4

Session b
Session #2
In this session you will help students connect their circuits and guide them
through the analysis of their data.
By using small time constants, we expect each measurement to fluctuate by
±5% from the theoretical value. It may be that some measurements don’t
fall within a standard deviations of the line of best fit, but the percentage
error should remain small.

We expect students to consider the possibility of internal impedance from the
PASCO unit. We expect them to propose possible methods to extract the
internal impedance of the PASCO. Testing showed that a rigorous experiment
to extract the internal impedance would be too time consuming for these lab
sessions.
Through testing, we found that the internal impedance of the PASCO is not
large enough to allow measurements of time constants of the order of 10s. To
remedy the situation, one can add a large impedance to the probes of the
PASCO. For more advanced students, push them to think of how the PASCO
and multimeters measure voltages and hence the limitations of the setup.
In this session, you are asked to build an RC circuit with a specific time constant. When
you enter the lab, each station will receive a unique circuit design and a given time constant.
Your circuit will have two sub-circuits: one that consists only of resistors and another one
that consists only of capacitors. You are to place both of these in series with the PASCO
source as shown in Fig. 4.8.

Figure 4.8: RC Circuit. You are to assemble sub-circuits that produces an equivalent
resistance Req and capacitance Ceq .
You will only use the “Positive Square Wave” input for this lab. Adjust them accordingly
to your calculated theoretical time constant for each measurement. Sometimes, the data
fitting tool cannot give standard deviations: this is because the data acquisition frequency
isn’t properly set. Change it until you see an uncertainty in the B parameter from the curve
fitting window.
You should test this subtlety in adjusting the acquisition frequency in order
to obtain the fit parameter.

There will be a few communal multimeters that allows to measure capacitance, while
every team will have their own multimeter that can only measure resistance. There will
also be boxes containing different resistors and capacitors. Their values will be labelled but
you can double-check their values using the multimeters. Note that we are using polarized
capacitors and therefore you must be careful when connecting them to the circuit: the voltage
input must be attached to a specific leg of the capacitor.
You should verify how polarized capacitors work, and explain to them the
proper way to connect them.
While every team will have to build a different RC circuit, the time constants that we are
asking are of the order of ms, and therefore you will have to use the ”Positive Square Wave”
input from the signal generator. You have the liberty to choose different values of frequency
depending on your estimate time constant. Changing the frequency of the input source does
not change the time constant: this only changes the duration that the source is ”turned on”.
Each team will have 5 capacitors of 10 ± 1µF and 5 resistors of 50 ± 1 Ω. Since there is a
limited quantity of multimeters that can measure capacitance, you can assume that these
standard deviations are correct. A summary of the equipment for this session is shown in
Table 4.1.
Individual station
Laptop
PASCO 850 interface
Resistance multimeter
Resistance and capacitance sub-circuit setups
Five 10 ± 1µF capacitors
Five 50 ± 1 Ω resistors

Collective station
Assorted resistors
Assorted capacitors
Capacitance multimeter

Table 4.1: Available material for Lab # 2
The goal of this lab is for you to convince us that you’ve constructed the right RC circuit
with the required time constant. You should discuss among yourselves and the TAs in order
to find appropriate tests to support your claim.
You must lead the discussion here. We want them to provide (at least) two
arguments. The first is to simply measure the effective resistance and capacitance of the circuit, and show a sample of their time constant measurement.

The second is to incrementally increase the resistance and capacitance separately. Doing so, and plotting τ as a function of the varying parameter,
they should find that the slope is the effective resistance or capacitance. This
provides an alternative approach to measuring the effective resistance and
capacitance of their setup, one that has more data points. The easiest approach is to connect resistors (the 50Ω ones) in series to the original effective
resistor, and capacitors in parallel (the 10µF ones) in parallel to the initial
capacitance.
Document your results and argue whether or not your RC circuit has the demanded time
constant. If you believe that it does, argue using standard data analysis methods. What is the
most appropriate measure to characterize the error of your measurements (see “Measurement
analysis” section of the “Basic statistics” section)? If your data doesn’t produce the correct
time constant, explain why. Explain what you must change in your circuit to remedy the
situation. Also discuss whether you believe this experiment is adequate to obtain RC time
constants. If possible, propose a better way to obtain time constants and/or decrease the
error in your experiment.

Chapter 5
Lab 3 - Magnetic Field of Solenoids
and Coils
5.1

Learning objectives

• Learn about solenoids.
• Learn how to use the right hand rule.
• Learn about vector fields.
• Learn about limits.
• Learn about electric motors.

5.2

Introduction

The goal of this lab will be to learn about real-world applications of magnetism. In particular,
we will study the solenoid in the first session, and the electric motor in the second. You
will use your data analysis skills previously acquired from the other labs to characterize the
magnetic field of these setups.
In the first session, you will be asked to draw the magnetic field lines of the solenoid
and to explain your findings in terms of the right hand rule. You will then quantitatively
characterize your solenoid: you must verify the dependence of the magnetic field B on the
current I through the solenoid and the the position z from the end of the solenoid on its
central axis.
In the second session, having learned how magnetic field strength varies with distance,
you will now use a permanent magnet to characterize an electric motor.

5.2.1

Pre-lab activity

1. How would you present data that consists of vectors?
Answer
Any vector data consists of information about its magnitude and orientation. This can be presented as a vector diagram, or tabulated into a
table.
2. What is a vector diagram? What needs to be present when drawing a vector diagram?
Answer
Magnitude and orientation. Typically, vectors are connected end-to-end, or
from a common origin.

3. Given the wire below, draw the field lines of the magnetic field generated from the
current in the wire.

Figure 5.1: Wire with a current I going from right to left.
4. Does the strength of the magnetic field increase, decrease or is it constant as we go
further away from the wire?
5. Imagine stacking multiple wires similar to that of Fig. 5.1 parallel to each other, with
the current of each all running in the same direction. Would the magnetic field close to
the stack of wires be greater, equal or smaller than the scenario with a single wire?
6. Given the wire below with a constant magnetic field B pointing downwards, what is
the direction of the magnetic force that B exerts on the wire?

Figure 5.2: A wire has a current running through it, hence electrons are running from right
to left. A constant magnetic field is present and points downwards.

5.3
5.3.1

Session a
Solenoid setup

In this session, you will use the IOLab that you previously used in PHYS 101 last semester
to measure magnetic field strengths.

Figure 5.3: Available solenoids. The one one the left has a smaller radius (1.9 cm radius, 550
turns, 15 cm of length, #19 gauge wire) [to be reverified], while the one on the right is larger
(3 cm radius, 560 turns and 15 cm of length, #16 gauge wire).
Each team will posses two solenoids as shown in Fig. 5.3. We will mostly use the one with
the smaller radius. You are to connect the two leads to the power supply. This will generate
a current through the wires.
Fig. 5.4 shows a typical power supply used for this lab. To use the power supply,
1. Make sure both knobs (red and green box) on the “master” side are completely turned
down.
43

Figure 5.4: Power supply.
2. Connect your wires in the orange box terminal (red for input, black for output).
3. Make sure to press down on both buttons in the blue box so that the power supply is
set on “parallel”. This will allow you to go up to 6A of current.
4. Turn the voltage knob to halfway (green box). This sets the maximum amount of
voltage allowed.
5. You will control the voltage and current of the power supply by slowly turning the
current knob (red box).

To gather data, you will use the IOLab:
1. Connect the IOLab USB to your laptop and open the IOLab software.
2. Calibrate your IOLab as shown in Fig. 5.5
3. On the left of the interface, select the “Magnetometer” and “Wheel” sensors. We will
only need the position data, not the velocity or acceleration plots.
4. When you are ready to acquire data, you can start recording by clicking the record
button. The data should appear as in Fig. 5.6.
The protocol is straightforward: you will move the IOLab to a given position and observe
the magnetic field strength at that position. Leave the IOLab at that position for a few
44

Figure 5.5: IOLab interface. Firstly, make sure the IOLab is connected to the bluetooth USB
stick: the area labelled by the orange box should indicate “Connected”. Afterwards, calibrate
your IOLab by clicking the button labelled by the red box → “Calibration” → “Remote
1” → “Accel - magn - gyro” and follow the subsequent on-screen instructions. We will use
the “Magnetometer” and “Wheel” sensors (see blue boxes). When you are ready to make
measurements, press the record button (purple box).
seconds so that you can select that interval in your data to obtain the standard deviation (σ)
and mean (µ) of that measurement. Repeat the above for all the positions that you wish to
measure the magnetic field.
Further comments about the IOLab can be found in the appendix.

5.3.2

Magnetic Field Lines
For this part of the lab, you will mostly help them understand the right hand
rule. Push them to think about the direction of the B field inside and outside
the solenoid, and its origin in terms of the right-hand rule.

Once the solenoid has been properly connected, you are asked to draw field lines on the
graph paper. For this, we will use the compass and IOLab. Place the compass on top of the
sensor. Note that the magnetometer isn’t centered in the IOLab; the magnetometer sensor of
the IOLab is located in one of the corner of the device as shown in Fig. 5.7.
First, chose at least a dozen points on the graph paper. For each point, you are to measure
the orientation of the magnetic field with the compass, draw a vector line associated with
your observation, and write next to the vector the measured strength of the the magnetic field.

Figure 5.6: Data sample. You can select an interval of your data. Doing so, you can obtain
directly the average (µ) and the standard deviation (σ) of your data within that interval. For
this lab, you should move the IOLab to a desired position, and wait a few seconds before
moving it again. This way, you can obtain an averaged value of the B-field at that position.
In the figure above, you would see a plateau. By doing this incrementally, your data should
look like a staircase as a function of time.
Recall that the total magnitude of the magnetic field will be given by Pythagoras theorem:
|B|=

Bx2 + By2 .

(5.1)

Both vector components can be obtained from the IOLab (see Fig. 5.6). Normally, one
would draw the length of the vector in a way that is representative of the magnitude of the
vector, however for the amount of vector lines you are asked to draw, we will simply write
the magnitude next to the drawn vector.
Once you have enough points, and have explored the possible orientations of the magnetic
field vector lines, answer the following questions in your log book:
1. Draw a diagram of the solenoid with the appropriate field lines.
2. What is the magnetic field inside the solenoid? Support your claim of the field lines
inside the solenoid in terms of the right hand rule.
3. What happens if you swap the connections of the solenoid? What would change?
Explain this in terms of the right hand rule.
4. Do these field lines look like any other magnetic object that you’ve seen? Explain why
this observation is true.
46

Figure 5.7: Position of the magnetometer on the IOLab (green box). Place the compass on
top at this position to obtain precisely the direction of the magnetic field detected by the
IOLab.

They should be reminded of the field lines of a magnet. Make them consider
why the field lines must loop around instead of simply diverging.

5. What parameters do you expect to affect the strength of the magnetic field of the
solenoid? Give a brief explanation for each parameter.
We expect them to
number of turns N .
physical motivation
them to verify the I

5.3.3

consider the current I, the distance z away, and the
We expect them to motivate the N dependence from a
(either increasing or decreasing with N ), but we want
and B dependence explicitly.

Current dependence

Now that you have a rough idea how a solenoid generates a magnetic field, we will quantitatively verify some of its possible parameters.
We will first verify its dependence on the current through the wires. Place the IOLab on
the ramp at a fixed distance away from one end of the solenoid (not too far) as shown in
Fig. 5.8, and increase the current through the wires with the power supply.
Verify how B depends on I using your data analysis skills from previous labs. Find the
parametric dependence on the current I and carefully support your claim by generating a
graph with an appropriate fit.

Figure 5.8: Measuring the distance dependence of the magnetic field. Place the ramp in front
of the solenoid and use it as a guide for the IOLab.

By taking increments of 0.5A, the data should be near perfect as the dependence was found to be linear with an approximate R2 ≈ 0.99 accuracy.

5.3.4

Distance dependence

We will now verify the magnetic field generated by a solenoid at a distance z (on the central
axis) away from the front of the solenoid.
Place the ramp in front of the solenoid such that the IOLab sensor is centred in front the
solenoid (see Fig. 5.8). Recall that the sensor isn’t centered. Move the IOLab incrementally,
and at each position, use the interface in order to obtain the average and standard deviation
of the position and magnetic field of the measurement. You will not need to go further than
30 cm. We recommend that you perform multiple trials to obtain many data points.
Once you have data for the magnetic field for all positions z < 30cm, we will first restrict
our analysis to the far z behaviour. We expect the data to show a 1/z 3 behaviour. Select
appropriate data points (equivalent to “far z”) such that you see a good 1/z 3 fit. Can you
explain the minimum value of z for which you performed a cutoff in your data? You may
perform this analysis again with the other solenoid if you wish.

There are many subtleties in this experiment. Firstly, they won’t be able to
find the correct 1/z 3 dependence without subtracting off background field
measurements: we found that surrounding electronics (laptop, phone, etc.)
will affect the magnitude of B. They must perform a run with and without
the B field of the solenoid turned on.
Having done that, through testing, we found that the far z behaviour corresponds to z > 4R, where R is the radius of the solenoid. We don’t expect
them to explain the factor of 4, but they should understand that it depends
on R, and it should be z R. Make them consider the consequences of the
near z behaviour, that is z → 0.
Why do you think that we find a 1/z 3 behaviour far away? Why doesn’t this fit work for
other regimes of z? Can you compare the 1/z 3 behaviour to other electromagnetic systems
that you may know: the electric field of a point charge, the magnetic field from a line ?
For most students, we expect them to find the 1/z 3 behaviour, and you are
to lead a discussion about why this fit only works for z R.
For the ambitious students, you can write down the equation of B for a solenoid
on the board, and they can try to compare their data to this theoretical curve.
Let the solenoid be centered at the origin z = 0, and zL , zR are the two ends
of the solenoid. Define L as the length of the solenoid L = zR − zL , N the
number of turns, I the current, then we have
dBz =

µ0 N I 2
dx
R
2L
((z − x)2 + R2 )3/2

µ0 N I
Bz =
2L

z − zL

z − zR

p
−p
(z − zL )2 + R2
(z − zR )2 + R2

µ0 N I R 2
1
.
Bz ≈
+
O
L z3
z5

(5.2)

In the last line, we expanded to leading order in the limit z → ∞. We want
them to see that taking appropriate limits, the equation simplifies greatly.
Since they don’t know calculus, you can show the first and last line of the
equation above which makes the relation obvious when z is large.

5.4

Session b

WARNING! The permanent magnets used in this experiment are quite strong. Avoid
bringing any magnetizable (made of iron/steel/Ni/Co, ID /credit cards) objects near the
magnets. Students with pacemakers could be excused from this lab.
In this session, we will study the electric motor. The electric motor is a device that
converts electrical energy into mechanical energy. Electric motors and their counter-parts,
electric generators, have been around for decades now and have become so ubiquitous in
our daily lives that life without them would seem almost impossible. Over the years, these
motors have grown in complexity and sophistication in order to perform a number of different
tasks but the basic physical principles behind their operation have not changed.
The goal of this lab is to provide a general understanding of how a simple DC motor
works with the hope that it might lead to a better appreciation for the more sophisticated
devices found all around us. This lab is not “complicated” to do, but will require a number
of different concepts in electricity, magnetism and circuit theory to complete.
The main observable of this lab is the frequency of rotation f of the motor. Similar to
the first capacitance lab, you are also asked to find possible parameters that could affect the
frequency of rotation of the motor.
We expect them to find that the the torque
τ = N ABI,

(5.3)

where N is the number of turns, A is the area of the loop, B is the magnetic
field and I is the current through the coil. They can’t vary the first two
parameters, but since the latter two are linearly dependent, the data analysis
should be familiar for them by now. Since this is the last lab session, you
should give them less advice as they should know how to analyse linear fits.

5.4.1

Electric motor setup

For this lab, you are given a power supply, a cathode ray oscilloscope (CRO), an electric
motor and a magnet. Fig. 5.9 shows the schematic diagram of the setup we will use in this
experiment and Fig. 5.10 shows a photograph of the actual setup.
To set up the apparatus:
• Make sure the power supply is turned off
• Connect the leads from the coil to the power supply.
50

Figure 5.9: Circuit diagram of the setup (front view)
• See the instructions for the CRO in the appendix.
The coil current and the voltage will be read from the power supply meters. The coil’s
period of rotation will be measured by a photodiode detector connected to a CRO. The
photodiode emits light and detects its reflection from the shiny strip covering one side of
the coil. The other side is covered by a non-reflective strip. When the coil is rotating, a
signal/pulse will be seen on the CRO screen every time the reflected light strikes the detector.
The period will be the time interval between two reflected signals/pulses. From the period you
can calculate the frequency. The performance of the motor and preciseness of measurements
are extremely sensitive to the positions of the photodiode and brushes. Please be careful
with the apparatus, and do not make any adjustments from how it was set up.
In addition to the electric motor, you are given a magnet on a track. Below (Fig. 5.11) is
a calibration of the magnet. This should remind you of your data in the last session.
The main concern for this lab is the equipment. It has been reported that
students tend to easily damage the apparatus and therefore you must be
vigilant of any deviations to the instructions.

Please respect the following instructions in order to avoid damaging the apparatus:
• Do not exceed a voltage of 10 V.
51

Figure 5.10: Photograph of the setup
• Do not let the magnet be any closer than 1 cm to the coils.
• It may happen that for a large enough voltage and close enough distance, the magnet
might vibrate or be pulled towards the coils. Do not let it be any closer than 1 cm to
the coils! Hold your magnet in place if necessary.
• If you are using high voltages (greater than 8 V), try to move quickly through the
higher voltages to reduce wear on the brushes.
• Remember to turn off the power supply after you have finished all your measurements.

5.4.2

Protocol

Similar to the first lab, you are asked to find what parameters affect the frequency of rotation
of the motor f .
To do this, recall that you must isolate your system and vary one parameter at a time in
order to perform the analysis. As a starting point, we suggest that you draw a force diagram
of the coils of the motor. Recall your answers from the pre-lab activity.
To make a measurement,
1. Make sure you read the previous instructions (above) in order to avoid damaging the
apparatus.
2. Set the magnet at a desired position (greater than 1 cm away from the coils).
3. Turn the power supply to a desired voltage (do not exceed 10 V).
52

Figure 5.11: B field as a function of distance from the face of the magnet.
4. The coil should start to rotate, though you will likely need to flick it gently with your
finger to get it started.
5. Measure the period T from the time interval between two successive pulses from the
CRO.
6. For better precision, you may count the time interval between a few pulses and divide
it by the number of intervals to get a better value for T . The frequency will be given
by f = 1/T . Note the value of f in an Excel sheet with any other relevant parameters
for this measurement.
Through discussions with your lab partners, other teams and the TA, you must perform
measurements to establish a (mathematical) relation between possible parameters that could
affect the frequency of rotation of the electric motor. Support your claims based on data and
figures as usual.

Part III
Appendix

Appendix A
Introduction to Excel
Firstly, every McGill student can obtain a free copy of Microsoft Offices software by clicking
here. Excel spreadsheet boxes can take in strings of letters of numbers as input. If the input
is a number, we can easily perform calculations.

Spreadsheet Calculations
• To initialize the math environment, simply click on a box, and press the “=” key on
your keyboard. Press “Enter” when you are done.
• One can perform basic addition (+), subtraction (-), multiplication (*) and division
(/) between two boxes. One can also take powers by using “ ˆ ” i.e. xˆ2 would be
x2 . First, press “=”, click on the box of your first variable, choose one of the above
manipulations and choose another box as your second variable.
• Excel obeys order of operators: multiplication and division come before addition and
subtraction. Parentheses are done first.
• Note that the middle bar above states the equality function in terms of boxes of the
spreadsheet.
• It is useful to separate your data into columns. In this way, say your first two columns
are your x and y variables that were obtained from a measurement. You should compute
f (xi , yi ) in another column. Doing so, after performing the mathematical manipulation
that you wish, you can click the bottom right green border of a box, and pull down to
the column to apply the same function along the columns of the input variables. In the
example of Fig. A.1, one could obtain the multiplication of each element of columns A
times those of column B displayed in column C.
• When defining a function with the intent of dragging the select-box down to apply the
whole function to all elements of columns, it is sometimes useful to define a box that
55

Figure A.1: Example of multiplication between two variables x and y. Notice that the selected
box has a green border around which the bottom right corner has an enlarged square.
doesn’t change as you drag the select-box. To do so, one uses the $ sign as in Fig. A.2.

Figure A.2: Fixing a box value when applying a function down a column. The $ sign before
the column letter means that we are fixing the column, and the $ sign before the row number
means that we are fixing the row. One can only use one $ sign to fix either the column or
row as desired.
• Excel also has functions that are included. Functions are called by some special
command in all caps. One typically calls a function, for example the sum, as
=SUM(B2:B5)
The colon means that we take every box between B2 and B5. One could have also used
that function by listing every box individually, separated by a coma. Alternatively, one
could, after writing “=SUM(”, selected the boxes that one wants to be included in the
function. This re-emphasizes the usefulness of arranging the data in columns.
• Other useful functions include:
– =STDEV.P(x) to calculate the standard deviation of a fluctuation population as
shown in eq. (2.6).
56

– =SQRT(x) to calculate square root of x.
– =Exp(x) to calculate exponentials ex .
– PI() to use the constant π. Note that there is no input.
– =Sin(x) and =Cos(x) to calculate values in radians. The input x must be in
radians.
– =AVERAGE(A2:A10) computes the average of the boxes between A2 and A10.
There exists many other functions that could be useful.

Figures
One can also create figures from data sets in Excel. To do so, select too columns and
go into the Insert tab as shown in Fig. A.3 to choose the type of figure that you with to
display. You can choose not-adjacent columns by selecting one column, press and hold Ctrl
while selecting the other column.

Figure A.3: Creating a plot based on some data. After selecting your data, go into the Insert
tab and choose the type of figure that you want. The red boxed symbol is the one for scatter
plots, which will most likely be your most used data display option.
Doing, so you should obtain something similar to Fig A.4. From here, one can do multiple
things.
• By clicking the green “ + ” symbol to the top right of the figure, you can add titles to
your axes, error bars, a legend and a trendline.
• By right-clicking the data points, you should see an option to “Format Data Series”.
Clicking that, you will see a right bar appear, and in the “Pain bucket” symbol, you
can change the size, shape and colors of your markers.
• By right-clicking a data point, you can add a trendline, in which case a right bar will
also appear. Here you can choose the type of equation that you wish to fit, and if
you scroll down completely, you will have the option to set an intercept, display the
equation on the chart, and display the R-squared value on the chart. You can also add
multiple trendlines to a single dataset to compare different fits.

Figure A.4: Scatter plot from data
• If you right-click an empty space on the grid of the chart, you can add multiple data
sets to a single chart by click “Select Data”. A pop-up window will appear with the
option of renaming how your datasets will appear on the legend, and the ability to
add/remove more datasets.
• Playing around with these options, you will have the ability to change the font size of
titles, colors of data points and texts, as well as change the thickness of the data points
and trendlines. You can move equations, legends or other things around your chart.
Don’t forget that data presented must be clear.
Finally, you can add data from .txt files. To do so, go into the data tab, and in the “Get
& Transform Data” section, there will be an option to add .txt and other types of files. Once
the popup screen to import data appears, you can click on “Edit” in the bottom right corner
to have more options to manipulate your data. A useful modification is to flip your rows and
columns. To do so, after clicking “Edit”, go into the “Transform” tab in the new window,
and select “Transpose”. Go back to the “Home” tab once you are done editing your file.

Appendix B
Lab Equipment
B.1

Multimeter

A multimeter allows one to measure various electronic properties. We will be mainly interested
in measuring the resistance or resistors, the capacitance of capacitors and the voltage and
current of a circuit. To use the multimeter, one must attach two wires into the input and
output. One can have alligator clips to attach to conducting wires more easily if necessary.
[insert pic of multimeter. Box certain useful settings]
To measure the resistance, simply take a resistor, turn the knob of the multimeter to the
resistance seting (Ω) and attach each wire to the two sides of a resistor. To measure the
capacitance of a capacitor, simply do the same as for the resistor, but turning the knob to
the capacitance setting.
To measure the voltage and current of a circuit,

B.2

Arduino

The Arduino unit requires the Arduino IDE software to run the desired scripts. Information
about the Arduino code for the capacitance lab can be found here.
The setup is schematically shown in Fig. B.1 where the example shows two resistors: one
with a resistance of 400Ω, the other is unspecified. We recommend using at least 106 Ω for
the unspecified resistor: you can choose different values to vary the time constant of the RC
circuit. This will change the precision of your setup.

Figure B.1: Arduino circuit scheme. The labels are associated with the pins and wire colors
of Fig. B.2.

Figure B.2: Corresponding real setup of the Arduino unit. The red boxes label the pins while
the purple boxes label the wiring that should make contact to each side of the capacitor.
To change the resistance of the resistor, you must also change the value in the Arduino
script. See Fig. B.1.

Figure B.3: Arduino resistor value. Change the value in the red box to the resistance value
of your setup. We suggest using at least 106 Ω.

B.3

Reading Resistors

Figure B.4: Resistor color chart.
The color code for resistors changes according to where the position (or number) of the
band is in the resistor, starting from where there is a greater number of bands. In the above
61

image this is from left to right. As a rule, the last two bands correspond to the multiplier
and tolerance respectively, while the rest correspond to the ohmic value. For example, as
shown in the figure, for the 4-Band-Code resistor we have: Green (5) and Blue (6), times 10
KΩ (Yellow) with a tolerance of ± 5% (Gold) so that resistor is 560 (± 5%) KΩ. For the
5-Band-Code resistor we have: Red (2), Orange (3) and Purple(7), times 1 Ω (Black) with a
tolerance of ± 1%(Brown) that is a resistor of 237 ±1% Ω.

B.4

Bread Boards

In Lab 2, you will use a bread board similar to that of Fig. B.5 to connect your RC circuit.

Figure B.5: A typical bread board used in Lab 2.
In the middle section of the bread board, there are metallic strips for every row (see for
example the green strips for rows 25-27 in the figure). On the side, there vertical strips as
shown for the red “+” column on the right. You may have another type of bread board, but
the design remains the same: you will have a section where metallic strips are distributed
horizontally, and another one where the strips are laid out vertically.

B.5

IOLab

Here are additional comments about the functionality of the IOLab
62

• You can zoom in and out
• How to get STD and mean
• How to load previous datasets

B.6

Oscilloscope

In lab 3, we will use a cathode ray oscilloscope (CRO). The CRO is a valuable tool often
used in many applications. Its most important function is the ability to measure voltages
that vary in time. The oscilloscope allows us to “visualize” how this voltage changes. To
fully characterize a voltage that varies in time using a hand-held voltmeter, you would have
to plot how the readings change with time on a graph. This is not only time consuming,
but impossible to do if the voltage changes too rapidly (for example, the voltage from a wall
socket changes sinusoidally from +115 V to -115 V and back to +115 V in 1/60th of a second).
The CRO report voltage by tracking the vertical position of an electron beam rapidly sticking
a fluorescent screen as it exists the space between two charged parallel conducting plates as
is illustrated in Fig. B.6. It plots the voltage on a screen along a vertical axis, and can also
track changes of this voltage in time (horizontal axis) in a fraction of a microsecond.

Figure B.6: Operation of a CRO
To use the oscilloscope you must first plug it into an outlet, turn it on, and then plug in
your positive and negative electrodes into the connection labeled Channel A (number 8 in
Figure B.7. You must then adjust the position (18), volts/div (12), and time/div (13) knobs
until you see a signal on the screen (19). The position knob will adjust the vertical position
of the signal. The volts/div and time/div on the other hand, will modifiy the value in volts
and time and in y and x respectively of each of the squares or divisions. Explained in another
63

manner these knobs allow you to control the zoom in/out in each direction separetely. The
oscilloscope has other options such as DC or AC coupling that will already be set up for you.
It will not be necessary to play around with any of the other options during the lab.

Figure B.7: Photograph of one CRO model

Appendix C
More statistics
Coefficient of determination
To determine the precision of a fit, one can look at its R2 value, also known as the
coefficient of determination. Consider a dataset of N datapoints with values yi , i going from
1 to N . We define the total sum of squares as
SStot =

N
X

(yi − ȳ)2 = (y1 − ȳ)2 + (y2 − ȳ)2 + ...(yN − ȳ)2 .

(C.1)

i=1

This is a measure of how your dataset differs from its average.
Now consider the values fi of a fit, where i are the values of the fit corresponding to xi of
yi value. We can also define the residual sum of squares which is a measure of how your fit
differs from your fit. This is given as
SSres =

N
X

(yi − fi )2 = (y1 − f1 )2 + (y2 − f2 )2 + ...(yN − fN )2 .

(C.2)

i=1

Finally, given the results above, we can define the coefficient of determination as
R2 = 1 −

SSres
SStot

(C.3)

which is a ratio of how your data fluctuations with respect to how you data differs from its fit.
A good coefficient of determination is R2 = 1. This can be computed automatically in Excel.

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