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Contents
Introduction
2
How to Write the Lab Report
5
Measurements and Uncertainties
7
Basic Electricity and Magnetism
11
Electric Fields and Equipotential Lines
15
Deflection of Electrons in an Electric Field
19
Ohm’s Law
23
Resistors in Series and Parallel
25
The Potentiometer
27
Determination of an Unknown Resistance Using a Wheatstone Bridge
31
Charging and Discharging of a Capacitor
35
Magnetic Field Around a Wire
37
Magnetic Field of Solenoid
39
Faraday's Law
41
Rays, Mirrors, and Thin Lenses
43
Diffraction of Light
47
Appendix A - Resistor Color Coding
49
Appendix B - The Breadboard
51
Appendix C - Uncertainty and Error Propagation
53
2014 Revisions: Nayeli Zuniga-Hansen, Ali Abu-Nada, Robert Baer, Richard West
1
Introduction
The purpose of the introductory physics laboratory is to give students the opportunity to
examine, firsthand, some of the concepts and laws of physics.
Physics is an accessible, tangible science. Physical phenomena are all around us at all
times, so much so that we often take the laws of physics for granted. In this lab you are asked to
examine more closely some natural phenomena and the laws that govern them. Physics is more
than abstract concepts and mathematical formulas; it is a dynamic, living field of study, which is
built upon information gathered from centuries of careful study and consideration of the
universe. The laws of physics are applicable to everything from the very large - planets, stars,
galaxies - to the very small - molecules, atoms, protons, and electrons. This course will show you
how scientists explore the physical laws of the universe.
The first step in our inquiry of the universe is to observe a physical entity or process. The
next step is to compare our observation with existing knowledge to see if the observation
conforms to our present understanding. If it does not then both theorists and experimentalists
examine the problem further. The theorist examines known information and then constructs a
model or theory within which the observation may be explained. The experimentalist gathers
further information to aid the theorist and also tests the predictions made by the theory. This
process may continue for decades as scientific knowledge is accumulated.
Most students who take introductory physics courses do not become scientists. However,
the principles, information, and techniques may be taken with them into any career. We live in
an increasingly technological society; as citizens we are being asked to make decisions which
will not only affect our lives, but the lives of many generations to come. A solid background in
scientific principles and an understanding of the way in which science is conducted will assist
you in making these difficult decisions.
Your objectives in this lab are as follows:
gain hands-on experience and firsthand knowledge of physical principles
acquire training in the scientific method including techniques of accurate observation
and the recording and handling of data
improve logical and rational thinking skills
acquire laboratory techniques including the operation and adjustment of specialized
equipment
understand the limitations and uncertainties inherent in scientific measurements
learn how to gather, organize, and analyze data to determine valid physical
relationships
learn how to present experimental conditions, observations, results and conclusions
Laboratory Instructions
Prior to each experiment, you should prepare as follows:
Read the entire experimental procedure as given in this manual.
Review the corresponding material in your text to gain a further understanding of the
physical concepts to be tested.
Prepare a "Prelab" as described on the next page.
2
Enter the lab with an understanding of the experiment and the procedure
The lab period is organized as follows:
1. The instructor will collect the Prelabs.
2. Students will go to a workstation and if necessary, make further preparations before the
class begins.
3. The instructor will discuss the experiment briefly and demonstrate the equipment.
Take note of any information the instructor may give that is not included in this manual,
especially precautions with regard to the proper handling of equipment. Some
apparatuses are especially fragile and easy damaged.
4. Students will then break into the smallest groups possible, usually two people, to carry
out the experiment. Cooperate with your partner(s) in such a way that each person has the
opportunity to use the experimental equipment. Work as quietly as possible so that others
may concentrate on their experiments.
5. Some equipment must be checked out from the instructor. One person from each group
will exchange a student ID card (or other appropriate ID) for the equipment. At the end of
the experiment the ID will be exchanged for the equipment.
6. Be honest in making and recording observations. Record data as it is indicated by the
equipment. If the results seem to be outside the limits of what is expected, recheck the
equipment and your calculations. If the result is still not what was expected, make the
best possible determination of the sources of error to explain the discrepancy.
7. At the end of the experiment, your work area must be cleaned and organized.
After the lab has been completed, prepare your Lab Report as described below. This report
should contain only your own work. Copy no data, calculations, or conclusions from any source
other than your own work.
Report Formats
The Prelab
This report is due at the beginning of the lab period. This report will count for a grade
and failure to submit can result in a zero grade for the experiment. After having studied this
manual and developed an understanding of the theory and procedure of the experiment, you will
prepare a one page report which will include the following: the title, your name, the object or
purpose of the experiment in your own words, a listing of the equipment you expect to use, a
paraphrased procedure, and a short (usually two or three sentences) description of the theory
behind the experiment.
The Lab Report
This report will be submitted after the experiment has been completed and will be due on
a date which will be specified by your instructor.
The final lab report includes:
1. Title Section: title, your partner's name, and the date the experiment was performed
3
2. Apparatus: a listing of the equipment which was actually used
3. Introduction: includes the theory and purpose of the experiment
4. Results: includes data sheets, calculations, data tables, and graphs
5. Error Analysis: includes error calculations and a discussion of the specific sources of
error
6. Discussion and Conclusion
7. Answers to Questions
The sections numbered 4, 5, 6 and 7 above are the most important sections of the report as they
demonstrate to your instructor your level of understanding of the experiment and these will be
weighed heavily in determining you grade. In section 5, error analysis, you will discuss the ways
in which the experimental results deviated from what had been predicted by theory. In the
discussion and conclusion section a serious statement is made about what was determined in the
experiment, the ways in which the experiment might be improved, and whether or not the tested
theory has been shown to be valid. At the end of the report all questions listed in the lab manual
and added by your instructor must be answered.
4
How to Write the Lab Report - Sample Instruction
Experimental Investigation of π
A sample lab report for this activity is provided below as an example for you to follow
when writing future lab reports.
Apparatus
Ruler, Vernier caliper, penny, marble, “D” cell, PVC cylinders.
Introduction
How is the circumference of a circle related to its diameter? In this lab, you design an
experiment to test a hypothesis about the geometry of circles. This activity is an introduction to
physics laboratory investigations. It is designed to give practice taking measurements, analyzing
data, and drawing inferences without requiring any special knowledge about physics.
Five objects were chosen such that measurements of their circumference and diameter
could be obtained easily and would be reproducible. Therefore, we did not use irregularly shaped
objects or ones that could be deformed when measured. The diameter of each of the five objects
was measured with either the ruler or caliper. The circumference and diameter of each object was
measured with the same measuring device in case the two instruments were not calibrated the
same. The circumference measurement was obtained by tightly wrapping a small piece of paper
around the object, marking the circumference on the paper with a pencil, and measuring this
distance with the ruler or caliper. The uncertainty specified with each measurement is based on
the precision of the measuring device and the experimenter’s estimated ability to make a reliable
measurement.
Results
The C/D value for the penny is (5.93 cm) / (1.90 cm) = 3.12 (no units). Results for all
five objects are given in the Table 1.
Table 1.
Object
Description
Diameter
± uncertainty
(mm)
Circumference
± uncertainty
(mm)
=C/D
(-)2
Penny
19.0±0.05
59.3±0.5
3.12
0.0004
“D” cell battery
33.0±0.05
104.5±0.5
3.17
0.0009
PVC cylinder A
42.3±0.05
133.0±0.5
3.14
0.00
PVC cylinder B
60.4±0.05
184.5±0.5
3.06
0.0064
Tomato soup can
66.0±0.05
212.0±0.5
3.21
0.0049
= 3.14
(-)2 = 0.0126
5
Error Analysis
The uncertainty of the instrument is ± ½ the smallest increment of measurement. If the
circumference is proportional to the diameter, we should get a straight line through the origin.
From our numerical results, we would expect the slope of the C vs. D graph to be equal to π. The
slope of the best fit line is 3.15, which is equal to π within its uncertainty. The R squared statistic
shows that the data all fall very close to the best fit line. If all the data lie exactly on the fitted
line, R squared is equal to 1. If the data are randomly scattered, R squared is zero. With an R 2
value of 0.997, our linear equation appears to fit the data very well.
Discussion and Conclusions
Our results support the original hypothesis for 5 circles ranging in size from 20 mm to 70
mm in diameter. The C/D ratio for our objects is essentially constant (3.14 ± 0.056) and equal to
π. The specified uncertainty is the standard deviation of the C/D ratio for the five objects.
Graphical analysis also supports the “directly proportional” hypothesis. The line has an intercept
(-0.5 ± 0.5) that is equal to zero within the uncertainty and a slope 3.15 equal to π within error. A
more extensive investigation of this C/D relationship over a wider range of circle sizes should be
performed to verify that this ratio is indeed constant for all circles.
Questions
1)
=[0.0126/(5-1)]1/2 = [0.00315]1/2 = 0.056
So, = 3.14 0.056
2) Plot the relation between c and d.
250
C = 3.15D - 0.502
Circumference
200
150
100
50
0
0
10
20
30
40
50
60
70
Diameter
C
𝐶 −𝐶
3) The slope = D= 𝐷2 −𝐷1 = 3.15.
2
1
4) The slope represents the numerical value of
𝐴𝑐𝑐𝑒𝑝𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝜋 − 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝜋
5) Percentage error = |
𝐴𝑐𝑐𝑒𝑝𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝜋
3.142 − 3.14
=|
3.142
| ×100%
| = 0.064 %
6
Experiment 1
Measurements and Uncertainties
Apparatus
Ruler, Vernier caliper, paper strip (or paper tape), five wooden discs of varying diameters
Introduction
Measurement
A measurement tells us about a property of something. It might tell us how heavy an
object is, or how hot it is, or how long it is. A measurement gives a number to that property.
Measurements are always made using an instrument of some kind. Rulers, stopwatches,
weighing scales, and thermometers are all measuring instruments. The result of a measurement is
normally in two parts: a number and a unit of measurement, e.g. ‘How long is it? ... 2 meters.’
Uncertainty of measurement
In an ideal world, measurements are always perfect: wooden boards can be cut to exactly
two meters in length and a known volume of steel will have a mass of exactly three kilograms.
However, we live in the real world and measurements are not perfect. In our world, measuring
devices have limitations. The imperfection inherent in all measurements is called an uncertainty.
In this laboratory, we will encounter uncertainty almost every time we make a measurement. Our
notation for measurements and their uncertainties takes the following form:
(Measured value ± Uncertainty) proper units
where the ± is read as “plus or minus”.
9.801 m/s2
9.794
9.796
9.798
9.800
9.802
9.804
9.806
Figure 1.1. Measurement and uncertainty: 9.801 ± 0.003 m/s2
Consider an acceleration due to gravity measurement, g = 9.801 ± 0.003 m/s2. We
interpret this measurement as meaning that the experimentally determined value of g can lie
anywhere between the values 9.801 + 0.003 m/s2 and 9.801 – 0.003 m/s2, or 9.798 m/s2 ≤ g ≤
9.804 m/s2. As you can see, a real world measurement is not merely one simple measured value,
but is actually a range of possible values (see Figure 1.1). This range is determined by the
uncertainty in the measurement. As uncertainty is reduced, this range is narrowed.
7
Sometimes we want to talk about measurements more generally, and so we write them
without actual numbers.
(X± ∆X)
and
(Y ± ∆Y)
In the laboratory, you will not only be taking measurements, but also comparing them.
You will compare your experimental measurements (i.e. the ones you find in lab) to some
theoretical, predicted, or standard measurements as well as to experimental measurements you
make during a second (or third) data run. We need a method to determine how closely these
measurements compare.
Standard deviation
Let's say we wanted to calculate the standard deviation for the amounts of gold coins
pirates on a pirate ship have. There are 5 pirates on the ship. In statistical terms this means we
have a population of 5 (N=5). If we know the amount of gold coins each of the 5 pirates have, we
use the standard deviation equation for a sample of population:
where, s = the standard deviation.
𝑥 = each value in the population.
𝑥̅ = the mean (average) of the values.
N= the number of values (the population)
In the case where we have a sample size of 5 pirates, we will be using the standard
deviation equation for a sample of a population. Here are the amounts of gold coins the 5 pirates
have:
x1=4, x2=2, x3=5, x4=8, x5=6.
Now, let's calculate the standard deviation:
1. Calculate the mean:
2. Calculate
for each value in the sample:
8
3. Calculate
4. Calculate the standard deviation:
Calculation of π
The equation of a straight line passing though the origin (x=0, y=0) is given by
y=mx
(1.1)
where, m is the slope of the line, x is the independent variable, and y is the dependent variable.
For any round object, the circumference (c) is directly proportional to its diameter (d).
cd
(1.2)
c = d (just like y = m x)
(1.3)
and
where is the constant of proportionality. By measuring the values of c and d of many round
objects and plotting c versus d, one can determine the value of from the slope of the line. You
will use a ruler and vernier caliper to measure the dimensions of c and d. Figure 1.2 shows how
you can read the vernier caliper.
9
Figure 1.2. Vernier caliper (Measurement Uncertainty and Probability, R. Willink)
The uncertainty of this caliper is ± ½ the smallest measurement, or ± 0.05 mm.
Procedure
Be careful when taking any of the measurements to keep errors as small as possible.
Measure the circumference of the given object by using the strip paper and the ruler. Then, use
the vernier caliper to measure the diameter of the five discs. Do this five times using five
different wooden discs. Record your measurements in Table 1.1.
Table 1.1.
c (mm)
d (mm)
=c/d
(-)2
1
2
3
4
5
=
(-)2 =
Questions
1) Calculate , where = (-)2/(N-1)1/2
2) Plot the relation between c and d.
3) Calculate the slope of the graph.
4) What does the slope here represent?
5) Calculate the relative error (see Appendix C), comparing your experimental value with the
theoretical value of = 3.142. Does this agree with your calculated uncertainty?
10
Experiment 2
Basic Electricity and Magnetism
Objectives
1) To study electric charges in a qualitative manner.
2) Explore the fundamental concepts of electrostatics.
3) Explore the fundamental concepts of magnetism.
4) Determine the magnetic field patterns around different types of magnets.
Note: Please, submit one lab report that contains the two parts.
Part One: Basic Electricity
Apparatus
Ebonite (gray) rod, clear rod, a piece of fur, a piece of paper, an electroscope
Introduction
There are two kinds of charges in nature, positive and negative. Positive charges are
carried by protons and negative charges are carried by electrons. When we say an object has a
charge on it, we mean that it has a slight excess of either positive or negative charge. For
instance, in this lab the ebonite rod will be given a negative charge by rubbing it with the fur. We
have not "created" more electrons on the ebonite rod, but rather, have moved some electrons
from the fur onto the rod. In so doing, the fur has excess protons and is positively charged. This
basic fact is commonly referred to as conservation of charge and is a fundamental concept of the
electromagnetic theory.
During this experiment, an electroscope (Figure 2.1) will be charged by two methods,
conduction and induction. To charge the electroscope negatively by conduction, the end of a
charged ebonite rod is touched to the ball on top of the electroscope and electrons flow from the
rod to the ball and the foil leaves, leaving a net negative charge. Because the leaves have like
charges on them, they repel from each other.
To charge the electroscope positively by conduction, the Lucite (clear) or glass rod is
rubbed with silk and touched to the ball on the electroscope. This time electrons flow out of the
electroscope onto the rod leaving a net positive charge on the electroscope foil leaves.
To charge the electroscope negatively by induction, a positively charged rod (Lucite
rubbed with silk) is brought near one side of the ball on top of the electroscope. At the same
time, touch the ball on the opposite side from the charged rod. The positively charged rod causes
a slight polarization on the ball, with electrons being attracted to the rod, leaving more positive
charges than negative on the side of the ball opposite the rod. When you touch the ball, electrons
flow from your finger to the ball, giving the electroscope a net negative charge. When the
charged rod is removed from near the ball, the leaves repel.
11
Figure 2.1. Electroscope (Elementary Lessons in Electricity and Magnetism, Sylvanus P.
Thompson (1881), MacMillan)
Procedure
This experiment is strictly qualitative and involves no calculations. This is, however, an
exercise in observation. You are expected to carefully follow the procedure and describe in
(reasonable) detail your observations. Any deviation from the described procedure will lead to
erroneous results.
Conduction
1) You will be given two plastic rods, a gray one and a clear one. Rub the gray plastic rod
with either the cotton or wool cloth in one direction. Note: If the rods are rubbed in a
back and forth manner they will not be properly charged. Charge the electroscope by
conduction using the gray rod. Draw a picture of the electroscope and the charged rod.
Assuming the charge on the gray rod is negative (indicated by a "-" sign) indicate on your
drawing the sign of the charge on the post and the sign of the charge on the plastic strip.
A positive charge is indicated by a "+" sign.
2) Rub the clear rod with paper, again in only one direction. Verify that the charge on the
post is the same sign as on the gray rod, but opposite to that on the clear rod. Describe the
method you used to carry out this verification.
3) Ground the electroscope to discharge it. You can do this by simply touching it. In this
case, you are ground. Repeat parts 1a and 1b using the clear strip and paper. In this
process you will again show that the charge on the clear strip is opposite to that on white
strip.
Induction
We will now use the charged clear rod to charge the electroscope by induction. Do the
following in the order given to avoid erroneous results.
1) Ground the electroscope before starting. Hold the clear rod near the post but not touching
it. Touch the post with your hand to ground it, and then remove your hand. Then, remove
the rod. Verify that the charge on the post is opposite to that on the clear rod and the same
as that on the charged clear rod. Describe the process you used to verify this. Draw
12
pictures to indicate the step-by-step process of induction. Be sure to indicate the sign of
the charge on the following: clear rod, metal foil, and the post.
2) Repeat part 1 using the gray rod instead of the clear rod. Report any broken equipment to
your instructor. Clean up your work area when finished.
Part Two: Basic Magnetism
Apparatus
Magnets, compass
Introduction
Certain elements and their alloys are observed to attract or repel like materials. This
attraction is not electrostatic in origin, but rather is due to a basic property of the atoms that make
up the material. The phenomenon we are referring to is called MAGNETISM. Every magnet,
regardless of its size or shape, has two unlike ends called magnetic poles. Because it was
observed that a suspended magnet will align itself with the earth's geographical north and south
poles, the unlike ends of the magnet are labeled NORTH and SOUTH.
The basic behavior of magnetic poles is similar to that observed for electric charges. Just
as electric charges gave rise to an electric field, two magnetic poles have field lines between
them (the magnetic field). Like poles repel each other while two unlike poles attract. The
direction of the lines is determined by the direction in which a north pole would travel when
exposed to a magnetic field. The field lines radiate outward from the north pole and enter the
south pole. A simple way of detecting magnetic fields around a magnet is to move a compass
around in its vicinity. The north pole of the compass will always point toward the south pole of a
magnet when it is brought close to the magnet.
Unlike the case of electric charges in which an individual charge (either positive or
negative) can be isolated, magnetic poles are always observed in pairs. If a magnet is cut into
pieces, each piece will have both a north and a south pole.
There are of course many sources of magnetic fields all around us. These not only include
magnetic materials but the earth itself, which acts like a giant magnet. Furthermore, as we will
see later, current carrying wires produce magnetic fields. Each of these sources contributes to the
magnitude and direction of the net magnetic field at any point.
Procedure
Basic Magnetic Properties
1) Place two bar magnets on your table and place all other magnets and metallic objects
away from the two bar magnets.
2) Note how the poles are labeled on one of the magnets. Pick up the other magnet, ignoring
the labels on it for now. Bring one of the poles near the north pole of the fixed magnet
without touching it and observe what happens. Record your observation in your
notebook.
3) Bring the same pole near the south pole of the fixed magnet and observe what happens.
Record your observation.
13
4) Turn the magnet in your hand around and use the opposite end to repeat parts 2 and 3.
5) Do the labels on the magnet in your hand and the behaviors you observed correspond to
the idea that like poles repel and unlike poles attract each other?
6) Take a ring magnet and place it on a pencil. Take another ring magnet and slide it down
the pencil toward the first magnet. Record your observation. Remove the second ring
magnet, flip it over, and again slide it down the pencil. Record your observation. Make a
sketch of each trial to illustrate your observations.
Mapping Magnetic Fields
1) Place a single bar magnet upon a sheet of graph paper. Remove all other magnets from
the vicinity of the bar magnet. Outline the shape of the magnet on the paper. Label the
north and south poles on your drawing.
2) Make a dot on your paper somewhere. This will be your starting point. Take a compass
and place it so that the needle points toward the dot. Mark the position of the other end of
the compass needle with a second dot. Move the compass beyond the second dot so that it
points at the second dot. Repeat this process until you have a line of points that either
ends at a pole or the edge of the paper.
3) Connect the series of dots with a smooth curve. This line represents a line of force and
should have arrows on it to indicate the direction of the magnetic field along the line.
4) Repeat step 2 until you have determined the general shape of the magnetic field around
the bar magnet.
Questions
Part one: Basic Electricity
1. When you comb your hair, you may have noticed a charge on your comb. What sign do
you think the charge on the comb is? How could you use your electroscope to verify
your answer?
2. Given the structure of the atom, explain why charge is usually transferred by electrons.
3. In this experiment, if you mistakenly removed the charged strip before you removed your
hand from the post, what would the result have been? Explain your answer.
Part two: Basic Magnetism:
1. Considering our discussion of magnetic fields, how is it that the north pole of a compass
needle points toward the earth's geographical North Pole?
2. If the earth is such a large magnet how can the magnetic field of such a small bar magnet
affect the position of a compass needle?
3. Can two different magnetic field lines ever cross each other? Why of why not?
4. How would you determine the polarities of the ring magnets? What would happen if the
pencil were removed while the ring magnets were positioned in each trial of step 6?
14
Experiment 3
Electric Fields and Equipotential Lines
Objectives
1)
2)
3)
4)
To map equipotential lines in an extended flat conductor traversed by an electric field.
To map the electric field lines.
To investigate electric field configurations based on the geometry of the electrodes.
To determine the intensity of the electric field (E) between two parallel plates.
Apparatus
Conducting sheets with electrodes (the first consisting of two parallel plates and the second
consisting of two circular electrodes), a voltmeter, a DC power supply, sheets of graph paper
Introduction
Gravitational fields are created by distributions of mass through space. In a similar
manner, static electric fields are created by charge distributions through space. The gravitational
field g at a point in space is defined to be equal to the gravitational force F acting on a test mass
m0 divided by the test mass. In other words, g = F/m0. An electric field can be defined similarly.
The electric field E at a point in space can be defined in terms of the electric force acting on a
test charge q0 placed at that point. The electric field at a point in space is defined as the electric
force acting on a positive test charge placed at that point divided by the magnitude of the test
charge.
F
(3.1)
E
q0
Note that E is the electric field external to the test charge and is not the field produced by the test
charge.
From the definition above we can see that the force on a positive charge due to an electric
field is in the direction of the electric field, while the force on a negative charge placed in the
same field is opposite to the direction of the electric field. Furthermore, the electric field is said
to exist regardless of whether or not a test charge is present. Once the electric field is known at
some point, the force on any charged particle placed at that point can be calculated. By
convention, we say that electric field lines exit from positive charges and enter negative charges.
For simple systems of high symmetry, the magnitude and direction of an electric field can be
determined using Coulomb’s law. When a more complicated system is introduced, these
calculations must be made numerically using complicated computer techniques. However, we
can determine the form of such complicated electric fields experimentally. In this experiment, we
will be doing just that. First, we will map the lines of equal potential, and then use these to
deduce the shape of the electric field between pairs of electrodes.
In order for a current to pass through a conductor, there must be a potential difference
between the two ends (or electrodes) of the conductor. This difference is a gradient between
these electrodes; i.e. there is a decrease in potential throughout the conductor from the point of
entry of electrons to the point of exit. This gradient is similar to the temperature change between
15
a heater and an open freezer a few meters away. The temperature is higher near the heater and
decreases, as we get closer to the freezer. The nearer we get to the entry point for the electrons,
the higher the potential.
In our experiment, we will not be finding an absolute potential, that is the potential with
respect to infinity, but we will determine the difference in potential between two points. We will
be using one electrode as a reference point. By finding a series of points that are all at the same
potential, we may connect these points and obtain an equipotential surface. Since we will only be
dealing with two dimensions, this surface is measured as an EQUIPOTENTIAL LINE. By
finding another series of points at the same potential, but different from our first series, we may
obtain another equipotential line and so on, until we have a series of lines that describes the
potential throughout the area between the electrodes.
Because there is no potential difference between any two points on the same
equipotential line, we may move a small test charge along this line without doing work. Our
definition of work is:
W Fd cos
(3.2)
where d is the distance moved along the line, F is the force, and is the angle between the
direction of movement and the direction of the force. Thus, if the work is zero, the force must be
perpendicular to the direction of movement. Restated, the electric field at each point along the
equipotential curve is normal to the curve. Since we can determine the shape of these curves, we
can, from these, determine the shape of the electric field in two dimensions.
To move a charge (Q) in an electric field along a linear path from point p1 to point p2, an
amount of work (W) must be done,
W12 = Q (V2 – V1)
(3.3)
where (V2 – V1) is the potential difference between the two points. The two points are said
to have the same potential if:
V2 – VR = V1 - VR
(3.4)
where VR is the potential of some reference point.
On conducting surfaces one would expect to find many such points that have the same
potential. These points are called equipotential points and the line joining them together is called
an equipotential line. Since the potential energy of the charge is the same at all these points, the
work done by the electric field on a charge moving along this line is zero. Therefore, the
equipotential lines must be at right angles to the electric field lines.
Procedure
1) Place the electrodes at the appropriate contact points on the conducting sheet.
2) On a graph paper draw a coordinate axis like the one on the conducting sheet and mark
the exact position of the electrodes.
3) Connect the circuit to a DC power supply of 6 volts, as shown in Figure 3.1. Use the
probe to find a point in the conducting sheet (x1, y1) which has the potential V1. Find four
16
more points (x2, y2), (x3, y3),…, which have the same potential as V1, and tabulate these
points in Table 3.1. Plot these points on the graph paper and connect them by a smooth
line. You have now found one equipotential line.
4) Choose four more locations for Vi and repeat the above procedure. Make sure that you
obtain a reasonable distribution of equipotential surfaces (for example use 1,2,3,4, and
5V) over the plane between the two electrodes and avoid clustering of the points.
5) Repeat steps 1-5 with the circular set of electrodes.
6) Plot the lines of force for each case.
V
V
(a)
(b)
Figure 3.1. Circuit diagram of the experiment.
Table 3.1.
Vi
(x1, y1)
(x2, y2)
(x3, y3)
(x4, y4)
(x5, y5)
V1
V2
V3
V4
V5
8) Connect the circuit shown in Figure 3.2 to measure the potential.
V
d
Figure 3.2. The circuit for measuring the magnitude of a uniform electric field.
17
9) Move the probe in regular steps in the area between the two electrodes and perpendicular
to them. Record the voltmeter reading and the corresponding distance from the negative
electrode in each step. You can use Table 3.2 below to tabulate your data.
10) Plot the voltmeter reading (V) versus the distance (d) and find the electric field (E).
Table 3.2. The measurements of voltage versus distance.
Trial
Voltmeter Reading
Distance from the negative electrode
number
V(volt)
d (m)
Questions
Part 1: Mapping of Equipotential Lines
1) Do the electric field lines cross? Can two equipotential lines ever cross? Explain.
2) Explain why the equipotential surfaces should be always perpendicular to the electric field
lines?
3) A uniform electric field is parallel to the y-axis. What direction can a charge be displaced
in this field without any external work being done on the charge?
Part 2: Determination of the Intensity of the Electric Field
1) Plot the voltmeter reading V versus the distance d.
2) Find the slope from this graph. How is it related to the electric field?
18
Experiment 4
Deflection of Electrons in an Electric Field
Objectives
To observe the effect of an electric field on a charged particle.
Apparatus
Cathode Ray Tube (CRT). Two power supplies:
a) The first is power supply model # 33033 which is used to create an accelerating
potential of 500 V and to supply a voltage for the heating filament.
b) The second power supply is used to produce an electric potential across the deflecting
plates.
Introduction
The study of the motion, or trajectory, of a charged particle through an electric field uses
principles from both classical mechanics and the electromagnetic theory. If an electron is placed
in an electric field it will experience a force. With the use of Newton’s Second Law, we can
express this force as:
F ma q E
(4.1)
Here F represents the force, a is the acceleration, m 9.1110 31 kg is the mass, and
q e 1.602 10 19 C is the charge of the electron.
This experiment utilizes a device called a CATHODE RAY TUBE (CRT). The cathode
ray tube is a device that is capable of emitting a beam of relatively fast moving electrons. The
name cathode ray was given to these emitted particles because physicist of the time had not yet
identified the particles as electrons.
Figure 4.1 (a), shows a simplified diagram of the CRT. The heater at the left is used to
heat the metal CATHODE where electrons are emitted from its surface. To the right of the
cathode is an ANODE with a small hole in the middle. The anode is held at a high positive
potential VA relative to the cathode. There is an electric field directed from the positive anode to
the cathode; therefore, the electrons are accelerated from left to right.
After leaving the anode, the electrons then pass between two parallel metal plates. These
plates deflect the electron beam and hence are called DEFLECTION PLATES. There are
actually two sets of these plates, one oriented horizontally and another set oriented vertically. To
understand how these plates deflect the electron beam, consider Figure 4.1 (b).
As the electron approaches the plates, its velocity is only in the horizontal direction, z. An
electric field is perpendicular to this direction and extends in the -y direction from the positively
charged plate toward the negative plate. As long as the electron is between the plates, it
experiences a force in the +y direction. The power supply provides a potential difference V DP
across the deflection plates.
19
(a)
(b)
Figure 4.1. Cathode Ray Tube (CRT).
The magnitude of the electric field between the plates is
E
VDP
d
(4.2)
where d is the distance between the plates. After the electron leaves the plates it no longer
experiences a force due to the electric field. With its acceleration now equal to zero the electron
continues along in a straight line toward the screen. A bright spot appears where the electrons
strike the screen. The position of the electron beam on the screen can be controlled by adjusting
the deflection plate voltage. The larger the potential difference across the deflection plates, the
greater the electric field and hence a greater deflection, s, is obtained.
In order to obtain an expression for the deflection of the electron beam we first need to
determine the velocity of the electron before it enters the plates. This can be obtained using the
conservation of energy:
1 2
1
mv0 qV A mvz2
(4.3)
2
2
where VA is the anode potential relative to the cathode, m is the electron mass, and vo and vz are
the initial and final velocities, respectively. Since the accelerating potential is very large, the
final velocity will be much greater than the initial velocity. Therefore, the initial velocity can be
approximated as being equal to zero. Solving for vz yields:
2qV A
2eV A
(4.4)
m
m
Here, e represents the magnitude of the charge of the electron. Once the electron enters the
region between the plates, it experiences acceleration in the y-direction.
E
ay e
(4.5)
m
We have used Newton's second law: F = ma = eE. The time required to travel the length L of the
plates is t = L/vz. During this time, the vertical velocity component becomes vy = ayt and the
electron is deflected a distance s given by
vz
20
2
1
1 eE L
s a y t 2
(4.6)
2
2 m v z
After leaving the plates, the electron continues toward the screen with a constant velocity
v v y2 v z2
(4.7)
Which makes the angle = tan-1(vy/vz) with the z-axis. The vertical deflection can now
be expressed as:
eE L 1 eE L
s D tan s D
m v z2 2 m v z
2
(4.8)
Substitution of the deflection plate potential from Equation (4.2) and the anode potentials
from Equation (4.4) into Equation (4.8) yields:
1
s
2d
L2 VDP
DL
2 VA
(4.9)
The term in brackets is a constant that depends entirely on the geometry of the CRT. If
the anode voltage is held constant, then a graph of s versus VDP should give a straight line with a
slope
1
L2 1
(4.10)
K DL
2 VA
2d
In this experiment, you'll create such a graph, be given values for L and D, and be asked
to determine an experimental value for d, the plate separation.
Procedure
Throughout this experiment you will be asked to observe the screen of the CRT. A grid
has been placed over the screen to allow you to measure the deflection of the beam as you adjust
VDP. When observing the screen, look at it straight on (not at an angle). The grid can be removed
and repositioned; therefore, once you begin your measurements do not move the grid. The major
grid lines are spaced 5 mm apart and the minor tick marks are spaced 1 mm apart. This
experiment utilizes a high voltage power supply. Exercise caution at all times.
1)
Two power supplies will be used in this experiment. The first is power supply model #
33033 which is used to create an accelerating potential of 500 V and to supply a voltage
for the heating filament. A 500 V potential difference is produced across the anode of the
CRT by using the –250 V and +250 V outlets of the power supply. The 6.3 V AC outputs
should be connected to the filament of the CRT. Once all connections to the first power
supply are verified by your TA, turn on the first power supply. Allow the filament to
warm up. If at any time you need to reconfigure the cables, turn off the power supply
before doing so.
21
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
The second power supply is used to produce an electric potential across the deflecting
plates. For the first part of this experiment the leads from the second power source should
be connected to the y-axis on the CRT. Once the proper connections have been made turn
on the power supply model # 33033 and let the CRT warm up.
When the CRT is properly warmed a small green dot should appear on your screen. This
image can be sharpened by adjusting the focus knob on the CRT. Note the position on the
screen where the dot appears. This is the position of the beam prior to being deflected.
Make certain that the variable voltage knob of the second power supply is set to its zero
position then turn on the power supply.
Slowly turn the variable voltage knob on the second power supply until a 3-volt reading
on the display is shown.
The image on the CRT should now have shifted from its original position. Record this
deflection as well as the voltage of the deflection plates in your notebook.
Repeat step (6) increasing the plate voltage in increments of 5 volts until a value of 30
volts is reached.
Reset the deflection plate voltage to zero then switch the positive and negative leads that
are connected to the deflection plates of the CRT.
Repeat steps (6) and (7) with the polarity of the plates reversed.
Set the deflection plate voltage back to zero then disconnect the wires from the y-axis
plates on the CRT and connect them to the x-axis plates on the CRT.
Repeat steps (6) through (9).
Construct a table for both the x and y deflection. Make certain that your table is properly
labeled and has the deflection plate voltages as well as the deflection distances of the
beam.
For each set of data, x-deflection and y-deflection, make a plot of s versus VDP. Then
draw a straight line through the data points such that equal numbers of data points are
above and below the line. Determine the slope, K, of the line and indicate the value.
Turn off both power supplies and reset the leads on the CRT to their original positions.
Using your values of K from each graph, calculate the deflection plate separation, d, for
each set of deflection plates. Note: Use the following values. D for the x-plates is 53mm.
D for the y-plates is 83 mm. L for the x-plates is 24mm. L for the y-plates is 21mm.
Questions
1. Our calculation of plate separation assumes simple flat parallel deflection plates. In practice,
the deflection plates actually have a flared end. Given this information, what do the measured
values of the plate separation distance that you calculated in step 15 actually represent?
Explain your reasoning.
2. A free electron and a free proton are placed in identical electric fields. Compare the electric
forces on each particle and discuss their subsequent accelerations.
3. Suppose the force of gravity on the electron was comparable to the electric force created by
the deflection plates. How would this change the results of your experiment?
22
Experiment 5
Ohm’s Law
Objective
To study the relation between the current flowing through a resistor and the potential drop
across its terminals.
Apparatus
Power supply producing an adjustable voltage from 0-30 V, two multimeters (to measure
voltage and current), a light bulb, two resistors, a breadboard, wires (for making connections)
Introduction
Part I: Breadboard Layout
A breadboard is used for prototyping circuits without the need for solder. It has many holes
(see appendix) that electrical components can be plugged into and connect with other
components. The holes are configured in rows and/or columns, and certain holes are “wired” to
each other by conductors under the holes. Using a multimeter as a continuity meter and two
jumper wires, explore how the breadboard is wired. Plug alligator-banana leads into the
multimeter: black to “COM”; red to “VΩ”, and turn the dial to the diode symbol
. The
multimeter will emit a tone when there is “continuity” indicating a circuit is complete. Touch the
two alligator clips together and make sure the multimeter emits a tone. Now, connect the
alligator leads to jumper wires, and probe the breadboard to determine which holes are
connected, and which are not connected. Draw a sketch of the breadboard indicating how the
holes are connected.
Part II: Ohm’s Law
When a potential difference is applied across a length of conducting material, an electric
field extends from one end of the conductor to the other. Freely moving electrons in the
conductor (conduction electrons) experience an electric force due to the electric field and move
in the direction opposite to that of the electric field. While this is what really happens, we use
conventional current in the study of circuits, where the current flows in the direction of the
electric field, which can be viewed as the motion of positive charges in the direction of
decreasing potential. This flow of charge is referred to as the current.
Ohm’s law states that if a conductor is kept at a constant temperature, the current (I)
flowing through it, is directly proportional to the potential difference (V) between its ends. That
is:
I = V/R
(5.1)
23
where R is the resistance of the conductor. The units of I, V and R in the SI system are Ampere,
Volt and Ohm, respectively. A conductor which obeys Ohm’s law is called an ohmic conductor.
Procedure
1) Set one of your multimeters to measure voltage (voltmeter) and the other to measure current
(ammeter).
2) Mount the unknown resistor provided by your instructor on your breadboard, and set up
the circuit below. Connect the voltmeter and the ammeter. The voltmeter is connected in
parallel with the resistor, while the ammeter is connected in series with the resistor.
V
R1
A
V
Figure 5.1. The circuit of Ohm’s law.
3) Vary the voltage of the power supply and obtain at least five pairs of readings of the
current (I) and the potential difference (V) across the resistor R1. Record these in a table.
4) Replace R1 by R2 and repeat. Record these in a table.
5) Plot two curves from your tables of current (I), as the independent variable, and
voltage (V) as the dependent variable on the same graph.
7) From the graphs determine R1, R2.
8) Replace the unknown resisters by a light bulb and repeat the experiment but take
12 data points between 0 and 12 V.
9) Compare between the graphs obtained using the resistors versus the light bulb.
Questions
1) What are the shapes of your graphs? Do you expect that they should be linear? If yes,
what does the slope of each one represent? Do they pass through the origin? Explain.
2) Find the values of R1 and R2 from the color codes (see appendix) and compare
(i.e. find the percentage difference) with your values.
3) Is the light bulb is an ohmic material? Explain why or why not.
24
Experiment 6
Resistors in Series and Parallel
Objectives
Investigate the series and parallel combinations of resistors.
Apparatus
Two resistors R1 and R2, multimeter, DC power supply
Introduction
Part 1: Resistors in series
In the first part of this experiment we will study the properties of resistors, which are
connected “in series”. Figure 6.1 shows two resistors connected in series (6.1) When two or more
resistors are connected together such that they have only one common point per pair of resistors,
they are said to be in series, as shown in Figure 6.1. The current I is the same through each
resistance. Since the potential drop from a to b equals IR1, and the potential drop from b to c
equals IR2, then the potential drop from a to c is:
V = IR1 + IR2 = I(R1 + R2) = IRequivalent
(6.1)
We can replace the two resistors by a single equivalent resistance Requivalent, whose value
is the sum of the individual resistances:
Requivalent = R1 + R2
(6.2)
V
a
R1
b
R2
c
I
A
V
Figure 6.1. Two resistors R1 and R2 connected in series.
In general, if n resistances are connected in series, then their equivalent resistance is:
Req R1 R2 R3 ..... Rn
(6.3)
Part 2: Resistors in parallel
Two or more resistors are said to be connected in parallel if they are connected as shown in
Figure 6.2. They have two common points per pair of resistors. In this case there is an equal
25
potential difference across each resistor, but the current in each resistor branch is generally not
the same.
The current I splits into I1 and I2 at the junction (point a) such that:
I = I1 + I2
I = (V/R1) + (V/R2) = V/Requivalent
(6.4)
(6.5)
From this result, we see that the equivalent resistance is given as:
1/Requivalent = (1/R1) + (1/R2)
(6.6)
In general, the equivalent resistance for n resistances connected in parallel is:
1/Requivalent = (1/R1) + (1/R2) + (1/R3) + … + (1/Rn)
(6.7)
V
I
a
I2
A
I1
R1
I
R2
Figure 6.2. Two resistors R1 and R2 connected in parallel
Procedure
1) Connect the two resistors R1 and R2 in series, and note that the ammeter is connected
in series with both resistors and power supply, while the voltmeter is connected in
parallel to both resistors (as shown in Figure 6.1).
2) Vary the voltage in the power supply and obtain at least five pairs of readings of the
current (I) and the potential difference (V) across the resistor Req.
3) Connect the two resistors R1 and R2 in parallel and repeat step (2) with I representing
the current in the circuit and V representing the potential drop across the common
points of R1 and R2. Record these in a table.
4) Plot two curves of current (I), as the independent variable, and voltage (V), as the
dependent variable, on the same graph.
5) From the graphs determine Req for series RS and parallel RP cases.
6) Compare your results of Req in both cases with their corresponding values found using
equations 6.2 and 6.6.
Questions
1) What conclusion can you draw, about the values of the total resistance in the case of
series connection (RS) and parallel connection (RP)?
2) A set of lamps are connected in series. What happens if one of them burns out? Explain
your answer.
3) Another set of lamps are connected in parallel. What happens if one of them burns
out? Explain your answer.
26
Experiment 7
The Potentiometer
Objectives
To determine the electromotive force, the terminal voltage, and the internal resistance of
a dry cell by the use of the potentiometer.
Apparatus
Slide wire potentiometer, galvanometer, DC power supply, resistor, connecting wires,
switch, voltmeter, dry cell, and secondary standard cell.
Introduction
A battery is a source of electric energy that can usually be obtained from two terminals
that have a potential difference. This difference is referred to as the TERMINAL VOLTAGE.
When current is running through the battery, the potential difference is decreased due to the
internal resistance of the battery. Thus, the maximum difference in potential usually occurs when
there is no current flowing through the battery. This maximum terminal voltage that the battery
can supply (if the internal resistance is negligible) is called the ELECTROMOTIVE FORCE, or
EMF of the battery.
It is often necessary to determine the EMF and the terminal voltage of a battery while
under load. A voltmeter is one of the more convenient methods used. However, this method is
not the most accurate. An ideal voltmeter would have an infinite resistance so that there would
be no current passing through it. Unfortunately, this is not possible; therefore, a small amount of
current will pass through the voltmeter. If the EMF of the battery is the desired quantity, then the
voltmeter can greatly affect the results.
In this experiment we will use an alternate technique, the potentiometer, to measure an
EMF. A potentiometer avoids the problems encountered when using a voltmeter. We will be
using a flow of charges (an electric current) and the EMF of two batteries. The device to balance
these EMF's is called a POTENTIOMETER. A potentiometer is shown in figure 7.1. It consists
of a steady power supply E, a slide wire AB, a rheostat Rh, a double-pole, double-throw switch
S, a resistance box R, a galvanometer G, a standard cell Es, and an unknown cell Ex. Since the
potential drop across a uniform wire is proportional to its length it can be shown that:
E x APx
E s APs
(7.1)
Thus if we know Es we can find APx and APs (the lengths of the wire used for the
unknown and standard cells, respectively) and we can calculate Ex, the EMF of the unknown
battery.
27
Figure 7.1. A schematic diagram of the potentiometer.
Procedure
It is extremely important that the circuit is wired precisely as the diagram states making
sure that the positive and negative terminals of the batteries and the power supply are attached as
shown (failure to do so will lead to erroneous or no results).
Part One: Calibration
1a)
Set the resistance box, R, to 10,000 ohms, the switch, S, to the standard cell and quickly
tap the sliding key to the wire at the 50 cm mark. Make certain that the sliding key
touches the wire for only a very brief period of time, less than a second. If the
galvanometer deflects, adjust the rheostat and tap again. If the galvanometer deflection is
larger, adjust the rheostat in the other direction and tap. Continue this process until no
deflection occurs in the galvanometer. This is called the balance point.
1b)
When the balance point has been determined, decrease R by 1,000 ohms and tap the key
against the wire at the 50cm point. If the galvanometer deflects, adjust the rheostat and
tap again. Repeat until there is no deflection.
1c)
Repeat step 1b until R is zero. After this point DO NOT CHANGE Rh. For a more
accurate balance gradually vary the location of the tapping of the slide key until there is
absolutely no deflection of the galvanometer.
1d)
Record the point at which there is no deflection as APs. The potentiometer is now
calibrated.
Part Two: Finding the Unknown EMF
2a)
Reset R at 10,000 ohms, and set switch S to the unknown cell.
28
2b)
Find the balance point by tapping the slide key at different points on the wire.
2c)
Reduce R by 1,000 ohms and tap the slide key at the balance point found in step 2b.
Gradually adjust the tapping location until a new balance point is found.
2d)
Repeat until R is zero. Record the final balance point as APx.
2e)
Using equation 7.1, calculate the EMF of the unknown cell.
Part Three: Terminal Voltage
NOTE: This is the first time that there will be current passing through the cell.
3a)
Connect the voltmeter and a resistor R=2 of the unknown cell in parallel as shown in
figure 7.2. Record the reading from the voltmeter. This is the terminal voltage of the
unknown cell.
V
R
Ex
Figure 7.2. Circuit for measuring terminal voltage of cell Ex using a voltmeter.
3b)
Now, replace the voltmeter with the potentiometer. Determine, using the potentiometer,
the potential difference between the terminals of the unknown cell. This is the true
terminal voltage. Record this voltage. Determine the percent difference between the two
values for the terminal voltage.
4b)
Using Ohm's Law, find the current through R. Since there is no current flowing through
the potentiometer, the only resistance used in this case is the known resistor. Kirchoff's
Loop Rule for this circuit is:
-Ir-IR=0
Here we have the EMF () measured in Part Two, the current I calculated in Part 3b,
R=2, and the internal resistance of the battery r. Use your experimental data to find the
internal resistance of the battery.
29
Questions
1. Is it essential that the power supply voltage (E) be greater than Ex? Discuss your answer.
2. Is it essential that the polarity of the power supply should be the same as Ex? Explain
your answer.
3. List the advantages and disadvantages of using a voltmeter to determine unknown
voltages. Do the same for the potentiometer. Which method is more accurate? Which
method is more practical? Which method is better? Explain.
30
Experiment 8
Determination of an Unknown Resistance Using a Wheatstone Bridge
Objective
In this experiment, we will study the fundamental properties of the Wheatstone bridge
circuit and how it is used to measure electrical resistances.
Apparatus
Wheatstone bridge, power supply, galvanometer, switch, resistance box, connecting
wires, and a resistor of an unknown resistance.
Introduction
The resistance of a conductor depends on the material of which it is made, its length, its
cross-sectional area, and its temperature. For constant temperature the resistance is given by
R
L
A
(8.1)
where R is the resistance in ohms, L is the length in meters, and A is the cross-sectional area in
square meters. The proportionality factor is known as the resistivity of the conductor and has
units of ohm-meter. Resistivity ( is constant for a given temperature and depends only on the
material used.
There are several different methods that can be used to measure the resistance of a
material, but one of the most accurate methods is the Wheatstone bridge. A Wheatstone bridge
consists of an unknown resistance, three known resistors, one of which is a calibrated variable
resistor, a galvanometer, and a source of emf. In principle the operation of the Wheatstone bridge
is quite simple. The variable resistor is changed until the galvanometer reads zero. At this point
the bridge is said to be balanced.
Due to the extreme sensitivity of
G
the galvanometer, the unknown
G
resistance can be measured very
X
1
accurately.
A
Wheatstone
bridge,
shown in Figure 8.1, is a circuit
2
3
that consists of the unknown
resistance (Rx), three known
resistors (R1, R2, and R3), a
3
2
galvanometer, a variable resistor
RG, and a power source. The thick
line represents the slide wire we
Figure 8.1. Setup.
will be using instead of two
31
individual resistors. To find the unknown resistance we must balance the circuit. When the
bridge is properly balanced, no current will flow through the galvanometer, i.e. IG = 0. It follows
from Ohm's Law that the voltage across the galvanometer (between points B and C) must be
zero. Hence the potential difference from A to either of the points B or C must be the same. That
is, the potential drop across Rx is equal to that across R2, so that
I s Rx I 2 R2
(8.2)
Similarly, the potential difference from B to D must be the same as that from C to D:
I 1 R1 I 3 R3
(8.3)
Since IG is zero, the current Ix would continue unchanged along the upper branch (i.e. Ix = I1) and
I2 would continue unchanged along the lower branch of the circuit (i.e. I2 = I3). Equations 8.2 and
8.3 become:
I x Rx I 2 R2
(8.4)
I x R1 I 2 R3
(8.5)
When equation 8.4 is divided by equation 8.5, we notice that the currents cancel, indicating that
their magnitudes do not need to be determined. The ratio of the resistances is
R x R2
R1 R 3
(8.6)
This is the balance condition for the bridge. Thus, if the value of the resistance R 1 is known and
the ratio R2 / R3 is known, the unknown resistance Rx may be measured. Because we are using
the slide wire form of the Wheatstone bridge (Figure 8.1) in this experiment the ratio may easily
be determined. A one-meter length of tantalum wire is connected between A and D to form the
lower branch of the circuit. A slicing contact divides the total resistance of the tantalum wire into
parts R2 and R3. Inspection of equation 8.1 shows the ratio R2/R3 is given by
R2 L2 / A L2
R3 L3 / A L3
(8.7)
The upper branch of the circuit consists of the known resistance R1 and unknown resistance Rx.
A decade substitution box will be used for R1. This type of resistor is adjustable to the nearest
0.1. The bridge can be balanced by adjusting the contact point C. The resistance of R x can be
found by inspection of equations (6) and (7).
The variable resistance RG is inserted in series with the galvanometer to protect it from
excessively large currents IG that may occur in initial balancing of the bridge. Balancing is done
by quickly tapping the slide wire while observing the galvanometer. R G is adjusted until the
bridge is balanced. Do not hold the sliding probe on the wire while balancing, just tap it quickly
and observe the galvanometer. Too much current through the galvanometer will damage it.
32
Procedure
This experiment does not require high voltage source. It is recommended that you use
approximately 5V. Check that the apparatus is set up as indicated in Figure 1. Set R x (the
Heathkit resistance substitution box) to 4.7 k. We begin with RG = 5 k; and as a more precise
balance is obtained, we will decrease it gradually to zero. An attempt should be made to balance
the bridge as close to the 50-cm mark along the slide wire as possible.
You will now determine the value of the unknown resistance Rx. Initially, the point of
contact C should be made at the 50 cm mark so that R2 = R3 (L2 = L3). R1 should be adjusted as
accurately as possible to obtain a balance point. Tap the wire for a fraction of a second and note
the amount of deflection of the galvanometer needle. Change the value of R1 and tap once again
to see if the galvanometer deflects the same way or the opposite way. Continue the procedure
until you succeed in finding two values of the resistance R1 that give oppositely directed
galvanometer deflections. The balance point must lie between these values. Remember, RG
should gradually be decreased as finer balancing is achieved.
When the adjustment of R1 is completed, vary the slice point C to obtain a still more
accurate balance. For your final balance, make RG = 0 by placing a lead across the terminals of
the resistance substitution box. Record the value of R1, L2, and L3. Calculate a value for the
unknown resistance. Be sure to reset RG = 5 k before measuring the next resistance.
1. Measure the resistance of the Heathkit resistance substitution box following the
procedure described above. Determine the percent error between the actual resistance
value and your experimental one.
2. Reset Rx (the Heathkit resistance substitution box) to 470 and repeat step 1.
3. Measure the resistance between the terminals 2 and 3, or 3 and 4, or 4 and 5, on the
unknown resistance spool as in Step 1. The type and length of wire of which the spool is
made is indicated. The actual resistance value of the spool can be determined by
obtaining values for the cross-sectional area A from you instructor, and by using
equation 8.1. Compare this to the value determined experimentally.
4. Report any broken equipment to your instructor. Clean up your work area when finished.
Questions
1. Why is it beneficial to balance the Wheatstone bridge as closely as possible to the 50-cm
mark along the slide wire? (Hint: The probe has a finite width and the accuracy of
measuring L2 and L3 depends on this width.)
2. To determine an unknown resistance, which is more accurate: Ohm’s law or Wheatstone
bridge method? Explain your answer.
3. Does it matter if the current supplied by the power supply is not constant?
33
34
Experiment 9
Charging and Discharging of a Capacitor
Objectives
1) To describe how the charge on a charging or discharging capacitor varies with time.
2) To state the correlation between the amount of charge on a capacitor and the voltage
across it.
3) To use technology to measure the voltage across the capacitor as it charges and
discharges.
4) To determine the time for the capacitor to charge to one-half of its maximum voltage.
5) Calculate the capacitance based on the time to 'half-max'.
Apparatus
USB Link, PASPORT Voltage-Current, Sensor, AC/DC Electronics Laboratory, 2 - 1.5
volt D-cell batteries
Introduction
Have you ever shocked someone by dragging your feet on a carpet and then touching the
person? As you drag your feet, you slowly accumulate charge. When you shock your friend, you
rapidly let go of the charge (in other words, you discharge). For a short time, you are a capacitor!
A capacitor is one of the most commonly used electronic circuit components. They can
store electric energy. Capacitors come in a variety of shapes and sizes, but they all behave in a
similar way. The voltage across a capacitor varies as it charges or discharges. Use a Voltage
Sensor to measure the voltage across a capacitor as it charges and discharges in a resistorcapacitor circuit. Use your data to calculate the capacitance of your capacitor. The value stamped
on the capacitor can be up to 20% different than the actual capacitance. Compare the calculated
value of the capacitor to the stated value of the capacitor.
Charging
Many electric circuits contain both resistors and capacitors. Figure 9.1 shows an example
of a resistor-capacitor or RC circuit. The charge on the plates builds up gradually to its
equilibrium value of qo = CVo, where Vo is the voltage of the battery. Assuming that the
capacitor is uncharged at time t = 0 s when the switch is closed, the capacitor charges
exponentially as shown in Equation 9.1.
q = 𝑞𝑜 (1 − 𝑒
−𝑡⁄
𝑅𝐶 )
(9.1)
The exponential e has the value of 2.718, q is the amount of charge at any time, qo is the
maximum charge achieved, t is the amount of time elapsed, R is the resistance of the circuit and
C is the value of the capacitor.
35
The term RC in the exponent in Equation 9.1 is called the time constant 𝜏 (tau) of the
circuit: 𝜏 = RC. The time constant is the amount of time required for the capacitor to accumulate
63.2% of its equilibrium charge. The charge approaches its equilibrium value rapidly when the
time constant is small and slowly when the time constant is large.
Figure 9.1. Schematic diagram for an RC circuit.
Discharging
Figure 9.2 shows a circuit just before a switch (S) is closed to allow a charged capacitor
to begin discharging. There is no battery in this circuit, so the charge +q on the top plate of the
capacitor (C) can flow counterclockwise through the resistor (R) and neutralize the charge -q on
the bottom plate.
Figure 9.2. Schematic circuit of the discharging process of a capacitor
The time constant (RC) is also the amount of time required for a charged capacitor to lose
63.2 % of its charge. The time it takes to charge (or discharge) the capacitor to half full is called
the half-life and is related to the capacitance and the resistance by Equation 9.2.
𝑡1/2 = 𝑅𝐶 𝑙𝑛2
𝑡1/2 = 𝜏 𝑙𝑛2
(9.2)
Procedure
This experiment is performed by using the computer in the Lab. Therefore, follow the
procedure which is written in the computer.
36
Experiment 10
Magnetic Field Around a Wire
Objectives
1) Gain some understanding of magnetic fields around a current-carrying wire.
2) Determine how magnetic field strength relates to current.
3) Find the permeability of free space and compare it to the accepted value.
Apparatus
2 USB Links, PASPORT Voltage-Current Sensor, PASPORT Magnetic Field Sensor,
Coil of wire, Patch Cords w/ alligator clips, DC Power Supply, Large Rod Base, Support Rod,
Buret Clamp
Introduction
A current-carrying wire experiences a magnetic force when placed in a magnetic field
that is produced by an external source, such as a permanent magnet. A current carrying wire also
produces a magnetic field of its own. Hans Christian Oersted (1777-1851) first discovered this
effect in 1820 when he observed that a current-carrying wire influenced the orientation of a
nearby compass needle. The compass needle aligns itself with the net magnetic field produced by
the current and the magnetic field of the earth. Oersted's discovery, which linked the motion of
electric charges with the creation of a magnetic field, marked the beginning of an important
discipline called electromagnetism.
If a current-carrying wire is bent into a circular loop, the magnetic field lines around the
loop have the pattern shown in the figure in the computer. At the center of a loop of radius R, the
magnetic field is perpendicular to the plane of the loop and has the value shown in equation 10.1,
where I is the current in the loop. Often, the loop consists of N turns of wire that are wound
sufficiently close together that they form a flat coil with a single loop. In this case, the magnetic
fields of the individual turns add together to give a net field that is N times greater than that of a
single loop. For such a coil, the magnetic field at the center is given by equation 10.2.
B=
μ0 I
2R
B=N
μ0 I
2R
(10.1)
(10.2)
Procedure
This experiment is performed by using the computer in the Lab. Therefore, follow the
procedure which is written in the computer.
37
38
Experiment 11
Magnetic Field of Solenoid
Objectives
1)
2)
3)
4)
To discover what the magnetic field is like inside a coil of wire known as a solenoid.
To calculate the magnetic field strength inside a solenoid.
To use a Magnetic Field Sensor to measure the magnetic field strength inside a solenoid.
To examine the relationship of the magnetic field strength to the position inside a
solenoid.
Apparatus
2 USB Link, PASPORT Magnetic Field Sensor, PASPORT Voltage-Current Sensor, DC
Power Supply, Primary and Secondary Coils, Patch Cords (Included w/ Sensor)
Introduction
A solenoid is a long coil of wire in the shape of a helix (see the figure in the computer). If
the wire is wound so the turns are packed close to each other and the solenoid is long compared
to its diameter, the magnetic field lines have the appearance shown in the drawing (see the next
page in the computer). Notice that the field inside the solenoid and away from its ends is nearly
constant in magnitude and directed parallel to the axis. The direction of the field inside the
solenoid is given by Right Hand Rule 2 (RHR-2), just as it is for a circular current loop.
The magnitude of the magnetic field in the interior of a long solenoid is shown in
equation 11.1. Where n is the number of turns per unit length of the solenoid, μ0 is the
permeability of free space, and I is the current.
B = μ0 𝑛𝐼
(11.1)
Procedure:
This experiment is performed by using the computer in the Lab. Therefore, follow the
procedure which is written in the computer.
39
40
Experiment 12
Faraday's Law
Objectives
1) Measure the voltage induced in a coil by a magnet that moves through the coil.
2) Determine the area under the curve of voltage versus time.
3) Compare the 'flux' (voltage x time) induced by one end of the magnet to the 'flux' induced
by the other end.
4) State whether or not the activity confirms Faraday's statement about voltage induced by a
magnet.
Apparatus
USB Link, PASPORT Voltage-Current Sensor, AC/DC Electronics Lab, Bar Magnet
Introduction
When electricity is passed through a conducting wire, a magnetic field can be detected
near the wire. Michael Faraday was one of the first scientists to reverse the process. The purpose
of this activity is to measure the electromotive force (emf) induced in a coil by a magnet
dropping through the center of a coil. Use the Voltage-Current Sensor to measure the voltage
induced in a coil as a bar magnet moves through the coil.
When a magnet is passed through a coil there is a changing magnetic flux through the
coil which induces an electromotive force (emf) in the coil. According to Faraday's Law of
Induction shown in equation 12.1, the emf, ε, depends on the number of coils, N, and the rate of
change of flux through the coils. In this activity, a plot of the emf versus time is made and the
area under the curve is found by integration. This area represents the flux as shown in equation
12.2.
∆φ
ε = −N
(12.1)
∆𝑡
ε∆t = −N∆φ
(12.2)
Procedure
This experiment is performed by using the computer in the Lab. Therefore, follow the
procedure which is written in the computer.
41
42
Experiment 13
Rays, Mirrors, and Thin Lenses
Objectives
To study the reflection and refraction of light for mirrors and lenses using the ray
approximation.
Apparatus
Ray box, 2 – Plane Mirrors with magnets attached, 2 – L bracket magnetic mounts,
Concave and convex metal mirror, Converging lens, Diverging lens, Plano convex lens, red
filter, green filter, Magnetic pad
Introduction
Light can exhibit both wave-like and particle-like properties depending on the way in which
light is used in an experiment. This experiment will focus on the wave properties of light. It may be
assumed for almost all-practical applications that light waves travel in straight lines. These straightline paths are known as RAYS. Figure 13.1 shows a light ray traveling in air from the left toward a
piece of glass. Part of the incident light is REFLECTED off the surface, i.e. it is as if it bounced off
the surface and travels away. The rest of the light enters the material but is deflected, or
REFRACTED, from its original path upon entering.
i
r
air
glass
t
Figure 1: Reflection and
Refraction of a Light Ray
Figure 13.1. Reflection and refraction of a light ray.
Through a series of studies we will examine the basic properties of light: refraction through
lenses and reflection from plane and spherical mirrors by following the paths of light rays. In this
experiment our source of light rays will be a ray box. The ray box is a simple incandescent light in a
box with slits on the front. Light emerges through slits so as to constitute a number of rays.
Normally, these rays diverge away from the source but they can be made to form a group of parallel
43
rays by proper placement of a plano-convex lens (see Figure 13.2). When only one or two of the
rays are needed the other slits on the box are covered with masking tape. (Important Note: Do not
place tape on any of the lenses, mirrors or filters used in this experiment.)
Figure 2
Figure 13.2. Plano-convex lens.
Procedure
When you are asked to trace along the edge of a mirror, always trace along the silvered side.
In the figures below, thick lines indicate the silvered side.
Part One: Law of Reflection
1a)
Place a plane mirror on a piece of paper. Set up the ray box such that a single ray exits the
box. Shine the ray towards the plane mirror at any angle i between 20 and 40 degrees as
shown in Figure 13.3. Trace on the paper the incident and reflected rays labeling them "i"
and "r," respectively. Before moving the mirror, trace a line along its silvered edge.
i
i
r
r
Figure 3: Reflection
From a Plane Mirror
Figure 13.3. Reflection from a plane mirror.
1b)
Now draw a perpendicular line to the line representing the silvered surface of the mirror.
1c)
Measure the angle i between the incident ray and the perpendicular. Measure the angle r
between the reflected ray and the perpendicular.
1d)
Compare your results with those predicted by the Law of Reflection, i.e., i =r.
44
Part Two: The Corner Cube
2a)
Stand two plane mirrors upright on a piece of paper. Form a right angle between them as
shown in figure 13.4. Note that the silvered sides face away from the incident light. Again,
use only one ray from the ray box. The mirrors should be placed so that the ray reflects off
of both mirrors.
Figure 4: A Top View
of the13.4.
Corner
Cube cube.
Figure
The corner
2b)
On the paper trace the incident and the two reflected rays, and the backsides of the mirrors.
2c)
Without moving the paper rotate the mirrors a few degrees and repeat step 2b. In your
notebook describe how the directions of the incident and reflected rays compare in each
trial.
2d)
Rotate the mirrors slowly by turning the paper beneath them. Describe how the direction of
the final reflected beam varies as the mirrors move.
Part Three: Converging Lens
3a)
Remove the masks from the ray box so that all of the rays may be used. Place the planoconvex lens close enough to the ray box so that the rays passing through the lens will
emerge parallel to each other.
3b)
Place the double convex lens on a piece of paper such that the light emerging from the
plano-convex lens enters the double convex lens. On the paper, trace incoming rays,
outgoing rays, and the outline of the double convex lens.
3c)
Measure the distance from the center of the traced lens to the point where the rays converge.
This distance is the FOCAL LENGTH, f, for this lens.
Part Four: Diverging Lens
Follow the same procedure as in Part Three, but replace the converging lens with a
diverging lens. For the diverging lens, you will need to extend the lines of the transmitted
rays back to a point in front of the lens to obtain the focal length.
45
Part Five: Reflection from Plane Mirrors
5a)
Place the red and green filters over the ray box slits such that the outermost rays are red and
green. The overlap of the red and green filters will black out the center rays. Note: Do not
tape the filters in place. The static formed between the filters and the ray-box should be
sufficient to hold the filters in place.
5b)
Place the plano-convex lens close enough to the ray box so as to produce parallel rays.
Arrange the two plane mirrors as shown in figure 13.5. Note the positions of the silvered
sides. When looking in the direction that the rays are propagating the green ray is on the
right. Place a piece of paper under the mirrors. Trace the paths of each ray for reflections off
each mirror.
Red Filter
Over Slit
Planoconvex lens
Ray
Box
Green Filter
Over Slit
Plane
Mirrors
Figure 13.5. Reflection from two plane mirrors.
Part Six: Concave and Convex Spherical Mirrors
6a)
Use all the rays from the ray-box and use the plano-convex lens to make them parallel. Place
the convex mirror in the path of the parallel rays. Trace the incoming rays, reflected rays,
and the reflecting surface of the mirror.
6b)
The focal length of the convex mirror is the distance from the center of the mirror to the
point of intersection of the reflected rays. Measure this distance.
6c)
Repeat the above procedure using the concave mirror. Extend the reflected rays behind the
mirror to find the focal point.
6d)
Report any broken equipment to your instructor. Clean up your work area when finished.
46
Experiment 14
Diffraction of Light
Objectives
1) To observe what happens when light passes through a double-slit interference pattern.
2) To observe what happens when light passes through a single-slit interference pattern.
3) To analyze both interference patterns to find the minimum and maximum intensities.
4) To compare the measurements of diffraction patterns
Apparatus
2 USB Link, PASPORT Light Sensor, PASPORT Rotary Motion, Sensor, 60 cm Optics
Track, Aperture Bracket, Laser Diode, Slit Accessory, Linear Translator
Introduction
In 1801, Thomas Young obtained evidence of the wave nature of light when he passed
light through two closely spaced slits. If light consists of tiny particles (or "corpuscles" as
described by Isaac Newton), we might expect to see two bright lines on a screen placed behind
the slits. Young observed a series of bright lines, and he was able to explain this result as a wave
interference phenomenon: the waves leaving the two small slits spread out from the edges of the
slits. In general, the distance between slits is very small compared to the distance from the slits
to the screen where the diffraction pattern is observed. The rays from the edges of the slits are
essentially parallel. For two slits, there should be several bright points (or "maxima") of
constructive interference on either side of a line that is perpendicular to the point directly
between the two slits.
The interference pattern for a single slit is similar to the pattern created by a double slit,
but the central maxima is measurably brighter than the maxima on either side. Compared to the
double-slit pattern, most of the light intensity is in the central maxima and very little is in the rest
of the pattern. As the width of the slit becomes smaller, the central diffraction maximum will be
the more intense. Equation 14.1 relates wavelength, the number of maxima, the slit width, and
the separation of the maxima in the bright fringes of a diffraction pattern.
sinθ =
nλ
𝑑
(14.1)
Procedure
This experiment is performed by using the computer in the Lab. Therefore, follow the
procedure which is written in the computer.
47
48
Appendix A - Resistor Color Coding
Black
Brown
Red
Orange
Yellow
Green
Blue
Violet
Gray
White
0
1
2
3
4
5
6
7
8
9
100
101
102
103
104
105
106
107
108
109
0
1
2
3
4
5
6
7
8
9
Color codes for the accuracy of a standard resistor: If the extra band is SILVER the
accuracy is 10%, for GOLD the accuracy is 5%.
Example: How to find the value of a resistance from the color codes. There are four color
strips; one of them is far from the other three. This one must be used to estimate the uncertainty
in the resistance.
In figure A.1, the first strip is yellow which corresponds to number 4 in the table above.
The second one is blue which corresponds to number 6. The third is green which corresponds to
number 5. The fourth strip is silver which corresponds to 10%. The accuracy in this resistance is:
ΔR = 4.6 10 6 10% 4.6 105 460 K . So, R ΔR = 4.6 10 6 4.6 105
Yellow
Blue
Green
Silver
Figure A.1. The value of this resistance is: R = 4.6 10 6 .
49
50
Appendix B - The Breadboard
Connection on Breadboard
The breadboard has many tiny sockets or holes arranged on a 0.1’’ grid. The leads or
terminals of most of the components like resistors, diodes, transistors, etc. can be pushed straight
into the holes to make a connection.
Letters are used on the breadboard (Figure B.1.) to identify horizontal rows and numbers
are used to identify vertical columns. The lines which are connecting the holes show how some
vertical columns and horizontal rows are internally connected. When a voltage is applied to the
breadboard current can flow along these internal connections.
Each column of five sockets in the inner sections is electrically connected to the others.
The two outer sections of the breadboard are usually used exclusively for power.
Figure B.1. The breadboard. (Safari Books)
51
52
Appendix C – Uncertainty and Error Propagation
In order to interpret the results of any qualitative measurement in an intelligent manner, an
understanding and thorough evaluation of all possible sources of error must be made.
ERROR is the difference between the true value of a physical quantity (such as mass, length,
time, etc.) and the best measured value. Errors are to be regarded as normal in any experiment,
but should be minimized to whatever extent possible. There any many different types of error
which you may encounter and these are defined and classified as follows. (I) Systematic (or
Persistent), (II) Random, and (III) Personal Errors.
I. Systematic Errors
These errors arise from three possible sources and are due to specific faults or limitations which
are inherent in the design of the experiment. These sources are classified as Method,
Environmental, and Instrumental errors.
A. Error Due to Method
One may have chosen a correct although inaccurate technique for making a given
measurement. You could measure the length of a room by waking toe-to-heel from one
side to the other and counting your footsteps. Then, you would multiply by the length of
your foot to get the distance. In practice, it would be difficult to measure the length
accurately in this way. A better way would be to use a steel tape measure or meter stick.
B. Environmental Errors
These are external factors such as temperature, humidity, vibration, wind, etc. which may
or may not be under the control of the experimenter and may affect the outcome of the
experiment.
C Instrumental Errors
These may arise from an instrument being used under conditions which are either beyond
its limitations or different from those under which it was calibrated. Another possibility is
that the instrument has been worn by use to the extent that accurate measurement can no
longer be made.
Systematic errors cause the experimental results to be either too high or too low in a
consistent manner throughout the experiment. The detection and elimination of systematic errors
is essential and a major consideration in quantitative experimental work.
II. Random Errors
Assuming an experiment has been set up to minimize or eliminate systematic errors, the
experimenter may notice that, even though all experimental conditions are identical for each
trial, the results are not identical for each trial. In fact; there may even be a large variation
between maximum and minimum values. this distribution of values is said to be due to random
53
errors. These errors are considered to be normal for any experiment. The magnitude of the error
is dependent on the nature of the experiment and the instruments being used.
A beginner might be tempted to eliminate accidental error by rounding off numbers until no
variations are indicated. The beginner might look at the data set (4.24, 4.22, 4.23, 4.22) and
round the values to all be identical (4.2, 4.2, 4.2, 4.2). The accidental variation has, thus, been
eliminated, but in doing so, a larger systematic error has been introduced. If a number of
independent readings agree exactly, the experimenter should, correctly, conclude that the
measuring instrument is capable of greater exactness than had been previously supposed.
Random errors usually fluctuate in magnitude and sign. They reflect the intrinsic precision of the
instruments and can be estimated theoretically by statistical methods. These errors may be
greatly reduced by repeating the measurement many times. As an example, consider the 11
readings of a distance listed in the table below. There are a range of values from 18.22 to 18.28
cm. Which value has the greatest probability of being correct? Statistics can show that the
arithmetic average is this number. The average is found by taken the sum of all the values (x)
and then dividing that sum by the number of observations (N). As a formula we write,
Arithmetic Average: 𝑥𝑎𝑣𝑒 =
∑𝑁
𝑖=1 𝑥𝑖
𝑁
where ∑ indicates the sum of values, and N is the total number of observations. For this
data, the average is 18.25 cm.
𝑥𝑖
(cm)
Observation Number, i
1
2
3
4
5
6
7
8
9
10
11
N – 1 = 10
18.26
18.25
18.28
18.24
18.25
18.23
18.22
18.25
18.27
18.26
18.24
∆𝑥𝑖2
(× 10 -4 cm2)
Deviation from xave
∆𝑥𝑖 (cm)
+0.01
0.00
+0.03
-0.01
0.00
-0.02
-0.03
0.00
+0.02
+0.01
-0.01
1
0
9
1
0
4
9
0
4
1
1
∑ ∆𝑥𝑖2 = 30 × 10-4
∑ 𝑥𝑖 = 200.75
Very often, experimenters are interested in determining the "spread" of their data: The
quantitative measure of how closely a set of data values come to the average is called the
standard deviation. The third and fourth columns in the table above will be used to calculate the
standard deviation.
Standard Deviation:
∑(𝑋𝑖 − 𝑋𝑎𝑣𝑒 )2
𝜎= √
𝑁−1
54
In the table, we have used the relation, ∆𝑥𝑖 = (𝑥𝑖 − 𝑥𝑎𝑣𝑒 ). In experiments where a large number
of observations are recorded, it is rare for individual readings to be more than 2 or 3 times the
standard deviation away from the average value. So, the standard deviation may be used to
represent the error for a set a data.
We can represent our data above as:
𝑥 = 𝑥𝑎𝑣𝑒 ± 𝜎
= 18.25 𝑐𝑚 ± 0.017 𝑐𝑚
Often, the experimental accuracy is indicated by calculating the percent deviation:
Percent Deviation = 100 (𝜎⁄𝑥𝑎𝑣𝑒 ) %
For our data the % deviation is 0.09 %.
Another measure of the error which is often used is RELATIVE ERROR. This is found by
comparing the measured values (Mexp) with theoretical values (Mth) and is usually given as a
percentage:
|𝑀𝑒𝑥𝑝 − 𝑀𝑡ℎ |
𝑅 = 100 [
]%
𝑀𝑡ℎ
where the lines in the numerator indicate that the absolute value is taken and Mexp and Mth
indicate the experimental and theoretical values for a measured quantity, respectfully.
III. Personal Errors
Personal errors generally arise from three causes: personal bias, judgmental differences, and
mistakes. Good experimentalists strive to recognize and eliminate these types of errors from
data.
A. Personal Bias
Scientific studies must be as objective as possible and eliminate the experimental beliefs
about what the outcome of a measurement should be. This bias is especially prevalent in
this lab course since we usually tell you what should happen theoretically in an
experiment and ask you to verify the prediction. This kind of error can prevent a faulty
theory from being detected and corrected or a new physical phenomenon from being
discovered. Students sometimes assume that the first of a series of measurements is
correct and then attempt to make subsequent measurement match the first one. This is not
correct procedure.
B. Judgmental Differences
In many types of measurements, such as reading a meter stick or a micrometer or some
other type of scale, the experimenter is required to observe and make a judgement about
55
the reading in order to record it. One person may tend to start a stop watch for a given
event later than some other person would. One way to determine if this type of error is
present is to have many people perform the same measurement.
C. Mistakes
Mistakes are errors which occur due to incorrect arithmetic, misplacement of a decimal
point, reversal of a sign, transposing digits, misreading an instrument, use of the wrong
scale, etc. Usually these can be detected during the experiment and do not play a part in
the final experimental results. One way to eliminate misreadings and wrong scale
readings is to simplify the experimental apparatus as much as possible. Taking great care
in making measurements and being aware of potential mistakes will eliminate these types
of errors in most cases.
Instrumentation Accuracy and Precision
The accuracy of an instrument is how close it measures to a true value such as a length, mass,
time, etc. The precision of an instrument is described by how repeatable the measurement is or
the amount of random error in the instrument. Accuracy of an instrument is generally obtained
by calibration, either when the instrument is built, or periodically as it is used. You will notice
throughout this lab that you have to “zero” a piece of equipment prior to taking data. This insures
that your data will be accurate. If you fail to zero an instrument, you will most likely see data
that is skewed largely one way or another having a very large systematic error. The precision of
an instrument is typically fixed with some exceptions. Electronic instrumentation with different
ranges will have increasingly higher precision as the range is lowered.
Instrumentation precision is either stated on the instrument, or assumed to be half the lowest
displayed value. In this way, the precision of a common meter stick could be said to be
± 0.5𝑚𝑚. This typical value of precision for a given instrument is appropriate for use when the
experimental setup is well suited to the measuring device and can be compared to standard
deviation of measurements in order to validate accuracy of data. In certain experiments it may be
obvious that the precision of the measurement device far exceeds that of the experimental setup.
In this case, you would expect the standard deviation of data to be much higher than the
precision of the instrument, and your total error should be reported accordingly.
Significant Figures
In recording observations is always necessary to be aware of the limitations of the measurement
and to take full advantage of the precision of an instrument. Thus, all digits which can be trusted
to be correct must be recorded even if the digits happen to be zero. For example, if a
measurement of the length of a piece of string can be made to the nearest 1/100 of a centimeter
and it is found to be 50.00 cm, the length should not be recorded as 50 cm since it implies that
the string is closer to 50 cm than it is to 49 an or 51 cm. Writing the length as 50.00 cm indicates
that it is between 49.99 and 50.01 cm. The uncertainty in the measurement is only 0.02%. When
we write 50.00 cm, we say that the fourth digit, counting from the left, is the last digit in which
we have confidence. The number of significant digits in this case is four. I£ the length were
written as 50 cm, this would be two significant digits.
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The number of significant figures is independent of the position of the decimal point. These
numbers all have three significant digits: 1.20 × 103, 4.00, 0.596, and 0.000220. If the number
contains both significant and non-significant zeros, this may be conveniently taken into account
by using scientific notation. For example, if 76000 is known to three significant figures, then
write 7.60 × 104. The following rules are used in the handling of significant figures:
Addition and Subtraction:
Do not carry the result beyond the first decimal place which contains doubtful numbers.
Example: 5.61 + 0.5715 + 12.176 = 18.3575 = 18.36.
5.61 only extends to the hundredths decimal place, therefore the result - should be
expressed to the nearest hundredth only. In dropping figures which are not significant, the
last digit which is retained is increased by one if the digit after it is greater than or equal
to 5.
Multiplication and Division:
Carry the result only to the same number of significant digits which the number in the
calculation with the least number of significant digits has.
Example: 45.73 × 2.56 = 117.0688 = 117.
45.73 has four significant digits while 2.56 has only three, therefore the result has three
significant digits.
Propagation of Error
When doing calculations on data with uncertainties in it, the uncertainties must be taken into
account. Typically this is done using either simple averages or standard deviations of errors. In
general, the relative errors add up to give a total error that is reported as: 𝑧 ± ∆𝑧
Basic Rules
Addition and Subtraction:
For an equation, z = x + y – z, uncertainty can be determined one of the following ways:
𝑢𝑠𝑖𝑛𝑔 𝑠𝑖𝑚𝑝𝑙𝑒 𝑎𝑣𝑒𝑟𝑎𝑔𝑒𝑠 𝑜𝑓 𝑒𝑟𝑟𝑜𝑟𝑠 ∶ ∆𝑧 = |∆𝑥| + |∆𝑦| + |∆𝑧|
𝑢𝑠𝑖𝑛𝑔 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛𝑠 𝑜𝑓 𝑒𝑟𝑟𝑜𝑟𝑠: ∆𝑧 = √(∆𝑥)2 + (∆𝑦)2 + (∆𝑧)2
Multiplication and Division:
For multiplication by an exact number, multiply the uncertainty by the same number:
𝑧 = 2𝑥
∆𝑧 = 2 ∆𝑥
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For numbers containing uncertainties:
𝑧 = 𝑥 𝑦 𝑜𝑟 𝑧 = 𝑥/𝑦
𝑢𝑠𝑖𝑛𝑔 𝑠𝑖𝑚𝑝𝑙𝑒 𝑎𝑣𝑒𝑟𝑎𝑔𝑒𝑠:
∆𝑧
∆𝑥 ∆𝑦
=
+
𝑧
𝑥
𝑦
∆𝑧
∆𝑥 2
∆𝑦 2
𝑢𝑠𝑖𝑛𝑔 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛𝑠
= √( ) + ( )
𝑧
𝑥
𝑦
Products of Powers
𝑧 = 𝑥𝑚 𝑦𝑛
𝑢𝑠𝑖𝑛𝑔 𝑠𝑖𝑚𝑝𝑙𝑒 𝑎𝑣𝑒𝑟𝑎𝑔𝑒𝑠:
∆𝑧
∆𝑥
∆𝑦
= |𝑚|
+ |𝑛|
𝑧
𝑥
𝑦
∆𝑧
𝑚 ∆𝑥 2
𝑛 ∆𝑦 2
√
𝑢𝑠𝑖𝑛𝑔 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛𝑠:
= (
) + (
)
𝑧
𝑥
𝑦
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