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University of Northern British
Columbia
Physics Program
Physics 206
Laboratory Manual
Winter 2012
ii
Contents
1 Measuring Wavelengths of Light Waves Using A Transmission Diffraction
Grating
5
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Measuring Mass-to-Charge Ratio e/m
2.1 Introduction . . . . . . . . . . . . . .
2.2 Experimental Procedure . . . . . . .
2.3 Supplementary Notes . . . . . . . . .
3 Michelson’s Interferometer
3.1 Introduction . . . . . . . . . .
3.2 Experimental Procedure . . .
3.2.1 Setup and alignment of
3.2.2 Measurements: . . . .
of
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the Electron
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the Interferometer:
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4 Determination of the Elementary Charge e Using Millikan Apparatus
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4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5 Photoelectric Effect
31
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6 Radioactivity
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 The Geiger-Mueller Counter . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.2 Random Nature of Radioactive Decay . . . . . . . . . . . . . . . . . . . .
6.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Determination of the Operating Voltage of the G-M Tube: . . . . . . . .
6.2.2 Statistics of Counting Experiments (Random Nature of Radioactive Decay)
iii
37
37
37
39
40
40
42
iv
7 Radioactivity II
7.1 Introduction . . . . . . . . . . . . . .
7.1.1 Inverse Square Law . . . . . .
7.1.2 Absorption of Radiation . . .
7.2 Experimental Procedure . . . . . . .
7.2.1 Inverse Square Law: . . . . .
7.2.2 Linear Absorption coefficient:
CONTENTS
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A Error Analysis
A.1 Measurements and Experimental Errors . . . .
A.1.1 Systematic Errors: . . . . . . . . . . .
A.1.2 Instrumental Uncertainties: . . . . . .
A.1.3 Statistical Errors: . . . . . . . . . . . .
A.2 Combining Statistical and Instrumental Errors
A.3 Propagation of Errors . . . . . . . . . . . . . .
A.3.1 Linear functions: . . . . . . . . . . . .
A.3.2 Addition and subtraction: . . . . . . .
A.3.3 Multiplication and division: . . . . . .
A.3.4 Exponents: . . . . . . . . . . . . . . .
A.3.5 Other frequently used functions: . . . .
A.4 Percent Difference: . . . . . . . . . . . . . . .
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45
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49
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55
55
List of Figures
1.1
1.2
Diffraction of light through a diffraction grating. . . . . . . . . . . . . . . . . . .
Diffraction Grating spectroscope. . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
2.2
2.3
e/m apparatus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Circuit diagram for the e/m tube. . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Circuit diagram for Holmholtz coils. . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1
Basic components of Michelson’s interferometer. . . . . . . . . . . . . . . . . . . 20
4.1
4.2
Circuit diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Various forces acting on the oil drop. . . . . . . . . . . . . . . . . . . . . . . . . 27
5.1
5.2
5.4
Circuit diagram for a photo cell. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Voltage vs current produced by light beams with the same frequency and different
intensities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Voltage vs current produced by light beams with the same frequency and different
intensities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Work functions of various photo cathodes. . . . . . . . . . . . . . . . . . . . . .
6.1
6.2
Circuit Diagram for a Geiger Mueller Counter. . . . . . . . . . . . . . . . . . . . 37
Voltage vs count rate of a G-M tube. . . . . . . . . . . . . . . . . . . . . . . . . 38
7.1
7.2
Geometry of a point source and a G-M tube. . . . . . . . . . . . . . . . . . . . . 46
Absorption of radiation in a slab. . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3
1
6
7
32
33
35
36
Introduction to Physics Labs
Lab Report Outline:
Physics 2??
section L?
date
Your Name
Student #
Lab Partner’s Name
Lab #, Lab Title
The best laboratory report is the shortest intelligible report containing ALL the necessary information. It should be divided into sections as described below.
Object - One or two sentences describing the aim of the experiment. (In ink)
Theory - The theory on which the experiment is based. This usually includes an equation
which is to be verified. Be sure to clearly define the variables used and explain the significance
of the equation. (In ink)
Apparatus - A brief description of the important parts and how they are related. In all but a
few cases, a diagram is essential, artistic talent is not necessary but it should be neat (use
a ruler) and clearly labelled. (All writing in ink, the diagram itself may be in pencil.)
Procedure - Describes the steps of the experiment. In particular, remember to state what
quantities are measured, how many times and what is done with the measurements (are they
tabulated, graphed, substituted into an equation, etc.). Then explain what sort of analysis will
be performed. (In ink)
Data - Show all your measured values and calculated results in tables. Show a detailed sample
calculation for each different calculation required in the lab. Graphs should be included on the
page following the related data table. (If experimental errors or uncertainties are to be taken
into account, the calculations should also be shown in this section.) Include a comparison of
experimental and theoretical results. (The graphs may be filled out in pencil but everything
else must be in ink.)
Discussion and Conclusion - What have you proven in this lab? Support all statements
with the relevant data. Was your objective achieved? Did your experiment agree with the
theory? With the equation? To what accuracy? Explain why or why not. Identify and discuss
possible sources of error. (In ink)
Introduction to Physics Labs
3
Lab Rules:
Please read the following rules carefully as each student is expected to be aware of and to
abide by them. They have been implemented to ensure fair treatment of all the students.
Missed Labs:
A student may miss a lab without penalty if due to illness(a doctor’s note is required) or emergency. As well, if an absence is expected due to a UNBC related time conflict (academic or
varsity), arrangements to make up the lab can be made with the Senior Laboratory Instructor
in advance. Students will not be allowed to attend sections, other than the one they are
registered in, without the express permission of the Senior Laboratory Instructor. Failure to
attend a lab for any other reason will result in a zero (0) being given for that lab.
Late Labs:
Labs will be due Tuesday at 3:00pm the week following the lab, after which labs will be
considered late. Late labs will be accepted up until 1 week from the beginning of the lab.
Students will be allowed 1 late lab without penalty after which all subsequent late labs will be
given a mark of zero (0). Any labs not received within one week will no longer be accepted for
grading and given a mark of zero (0).
Completing Labs:
It is unlikely that these lab reports can be completed within the scheduled lab time. Given
that the lab time is Friday in the late afternoon it is highly recommended you come prepared
prior to the lab. The due date is Tuesday so as to allow adequate time for questions on the
Monday. Please allot sufficient time over the weekend to work on the lab in case you need to
ask questions on Monday.
Conduct:
Students are expected to treat each other and the instructor with proper respect at all times
and horseplay will not be tolerated. This is for your comfort and safety, as well as your fellow
students’. Part of your lab mark will be dependent on your lab conduct.
4
Introduction to Physics Labs
Plagiarism:
Plagiarism is strictly forbidden! Copying sections from the lab manual or another person’s
report will result in severe penalties. Some students find it helpful to read a section, close the
lab manual and then think about what they have read before beginning to write out what they
understand in their own words.
Use Ink:
The lab report, except for diagrams and graphs, must be written in ink, this includes filling
out tables. If your make a mistake simply cross it out with a single, straight line and write the
correction below. Professional scientists use this method to ensure that they have an accurate
record of their work and so that they do not discard data which could prove to be useful after
all. For this reason also, you should not have “rough” and “good” copies of your work. Neatly
record everything as you go and hand it all in. If extensive mistakes are made when filling out
a table, mark a line through it and rewrite the entire table.
Lab Presentation:
Marks will be given for presentation, therefore neatness and following the outline is important.
It makes your lab easier to understand and your report will be evaluated accordingly. It is
very likely that, if the person marking your report has to search for information or results, or
is unable to read what you have written, your mark will be less than it could be!
Contact Information:
If your lab instructor is unavailable and you require extra help, or if you need to speak about
the labs for any reason, the Senior Laboratory Instructor, Jon Schneider, is available by email:
[email protected] or phone: 250.960.5708. Office hours are posted on the physics lab webpage
www.unbc.ca/physics/labs.html.
Supplies needed:
Clear plastic 30 cm ruler
Metric graph paper (1 mm divisions)
Paper duotang
Lined paper
Calculator
Pen, pencil and eraser
Experiment 1
Measuring Wavelengths of Light Waves
Using A Transmission Diffraction
Grating
1.1
Introduction
In one Physics 202 experiment the prism spectrometer had to be calibrated in terms of known
wavelengths before we were able to determine unknown wavelengths of light experimentally.
How, then, are the wavelengths of spectral lines or colors initially determined? This is most
commonly done with a diffraction grating, a simple device that allows for the study of spectra
and the measurement of wavelengths.
By replacing the prism with a diffraction grating, a prism spectrometer becomes a grating
spectrometer. When using a diffraction grating, the angle(s) at which the incident beam is
diffracted relate directly to the wavelength(s) of the light. In this experiment the properties of
a transmission grating will be investigated and the wavelengths of several spectral lines will be
determined.
A diffraction grating consists of a piece of metal or glass with a very large number of evenly
spaced parallel lines or grooves. This gives two types of gratings: reflection gratings and transmission gratings. Reflection gratings are ruled on polished metal surfaces; light is reflected from
the unruled areas, which act as a row of “slits.” Transmission gratings are ruled on glass, and
the unruled slit areas transmit incident light.
The transmission type is used in this experiment. Common laboratory gratings have 300 grooves
per mm and 600 grooves per mm (about 7500 grooves per in. and 15,000 grooves per in.), and
are pressed plastic replicas mounted on glass. Glass originals are very expensive.
6
EXPERIMENT 1
Figure 1.1: Diffraction of light through a diffraction grating.
Diffraction refers to the “bending” or deviation of waves around sharp edges or corners. The
slits of a grating give rise to diffraction, and the diffracted light interferes so as to set up
interference patterns (Figure (1.1)). Complete constructive interference of the waves occurs
when the phase or path difference is equal to one wavelength, and the first-order maximum
occurs for
d sin(θ1 ) = λ
(1.1)
where d is the grating constant or distance between the grating lines, θ1 the angle the rays are
diffracted from the incident direction, and d sin θ1 the path difference between adjacent rays.
The grating constant is given by:
d=
1
N
(1.2)
where N is the number of lines or grooves per unit length (usually per millimeter or per inch)
of the grating.
A second-order maximum occurs when d sin θ2 = 2λ, and so on, so that in general we may
write:
d sin(θn ) = nλ
n = 1, 2, 3, ....
(1.3)
where n is the order of the image maximum. The interference is symmetric on either side
of an undeviated and undiffracted central maximum of the slit image, so the angle between
symmetric image orders is 2θn (Figure (1.2)).
MEASURING WAVELENGTHS OF LIGHT WAVES USING A TRANSMISSION
DIFFRACTION GRATING
7
Figure 1.2: Diffraction Grating spectroscope.
In practice, only the first few orders are easily observed, with the number of orders depending on the grating constant. If the incident light is other than monochromatic, each order
corresponds to a spectrum. (That is, the grating spreads the light out into a spectrum.) As
can be seen from Equation (1.1), since d is constant, each wavelength (color) is deviated by
a slightly different angle, so that the component wavelengths are separated into a spectrum.
Each diffraction order in this case corresponds to a spectrum order. (The colorful displays seen
on compact disks [CDs] result from diffraction.)
1.2
Experimental Procedure
1. Review the general operation of a spectrometer if necessary. Record the number of lines
per metre of your diffraction grating . Mount the grating on the spectrometer table with
the grating ruling parallel to the collimator slit and the plane of the grating perpendicular
to the collimator axis.
DETERMINATION OF THE WAVELENGTH RANGE OF THE VISIBLE SPECTRUM
2. Mount an incandescent light source in front of the collimator slit. Move the spectrometer
telescope into the line of the slit of the collimator and focus the cross hairs on the central
slit image. You will want to adjust the scale so that the angle reading is zero at this
point. Notice that this central maximum or “zeroth”-order image does not depend on
the wavelength of light, so that a white image is observed. Then move the telescope to
either side of the incident beam and observe the first- and second-order spectra. Record
which one is spread out more in you laboratory report.
8
EXPERIMENT 1
3. Focus the cross hairs on the blue (violet) end of the first-order spectrum at the position
where you judge the spectrum just becomes visible. Record the divided circle readings
(to the nearest minute of arc).
4. Move the telescope to the other (red) end of the spectrum and record the divided circle
reading of its visible limit.
5. Repeat this procedure for the first-order spectrum on the opposite side of the central
maximum. The angular difference between the respective readings corresponds to an angle of 2θ (Figure (1.2)).
6. Compute the grating constant d in millimeters and with the experimentally measured θ0 s
compute the wavelengths of red and blue ends to define the range of the visible spectrum
in nanometers and angstrom units.
DETERMINATION OF THE WAVELENGTHS OF SPECTRAL LINES
7. Mount the mercury discharge tube in its power supply holder and place in front of the
collimator slit.
Caution: Work very carefully, as the discharge tube operates at high voltage and you
could receive an electrical shock.
Turn on the power supply and observe the first- and second-order mercury line spectra
on both sides of the central image.
8. Because some of the lines are brighter than others and the weaker lines are difficult to
observe in the second-order spectra, the wavelengths of only the brightest lines will be
determined. Find the listing of the mercury spectral lines in Table (1.1), and record the
color and wavelength.
Then, beginning with either first-order spectra, set the telescope cross hairs on each of
the four brightest spectral lines and record the divided circle readings (read to the nearest
minute of arc). Repeat the readings for the first-order spectrum on the opposite side of
the central image and find the average angle and its error for each line.
9. Repeat the measurement procedure for the four lines in the second-order spectra and,
using Equation (1.1), compute the wavelength of each of the lines for both orders of
spectra. Compare with the accepted values by computing the percent error of your
measurements in each case. Which order gives you more accurate results?
Note: In the second-order spectra, two yellow lines, a doublet, may be observed. Make
certain that you choose the appropriate line. (Hint: See the wavelengths of the yellow
lines Table (1.1). Which is closer to the red end of the spectrum?)
MEASURING WAVELENGTHS OF LIGHT WAVES USING A TRANSMISSION
DIFFRACTION GRATING
Table 1.1: Wavelength of visible spectral lines of Helium and Mercury.
Wavelength
Element (nm) (Å)
Helium
388.9 3889
396.5 3965
402.6 4026
438.8 4388
447.1 4471
471.3 4713
492.2 4922
501.5 5015
587.6 5876
667.8 6678
706.5 7065
Mercury 404.7 4047
407.8 4078
435.8 4358
491.6 4916
546.1 5461
577.0 5770
579.0 5790
690.7 6907
Color
Relative Intensity
Violet
1000
Violet
50
Violet
70
Blue-Violet
30
Dark Blue
100
Blue
40
Blue-Green
50
Green
100
Yellow
1000
Red
100
Red
70
Violet
300
Violet
150
Blue
500
Blue-Green
50
Green
2000
Yellow
200
Yellow
1000
Red
125
9
10
EXPERIMENT 1
Experiment 2
Measuring Mass-to-Charge Ratio e/m
of the Electron
2.1
Introduction
When a charged particle such as an electron moves in a magnetic field in a direction at right
angles to the field, it is acted on by a force, the value of which is given by
f = Bev
(2.1)
where B is the magnetic-flux density in webers/meter2 (W b/m2 ), e is the charge on the electron
in coulombs (C), and v is the velocity of the electron in meters/sec (m/s). This force causes
the particle to move in a circle in a plane perpendicular to the magnetic field. The radius of
this circle is such that the required centripetal force is furnished by the force exerted on the
particle by the magnetic field. Therefore
mv 2
= Bev
r
(2.2)
where m is the mass of electron in kg, and r is the radius of circle in m, i.e. 1/2 D. If the
velocity of the electron is due to its being accelerated through a potential difference V , it has,
due to its velocity, a kinetic energy of
1 2
mv = eV
2
(2.3)
where V is the accelerating potential in volts. Substituting the value of v from Equation (2.2)
into Equation (2.3), we get:
e
2V
= 2 2
m
B r
(2.4)
12
EXPERIMENT 2
Thus when the accelerating potential, the flux density of the magnetic field, and the radius of
the circular path described by the electron beam are known, the value of e/m can be computed,
and is given in C/kg by Equation (2.4) if V is in volts, B is in W b/m2 , and r is in m.
The magnetic field which causes the electron beam to move in a circular path is produced by
Helmholtz coils. The magnetic field produced by Helmholtz coils can be expressed, in terms of
the current going through the coils and the dimensions of the coils, as:
8µ◦ N I
B= √
125a
(2.5)
where:
N is the number of turns of wire in each coil,
I is the current through the coils in amperes (A),
a is the mean radius of the coil in meters,
µ◦ is the permeability of empty space, which is 4π × 10−7 W b/A · m.
Substituting Equation (2.5) in Equation (2.4) gives
3.91 a2 V
e
= ( 2 · 2) 2 2
m
µ◦ N I r
e
a2 V
= (2.47 × 1012 2 ) 2 2
m
N I r
(2.6)
Equation (2.6) is the working equation for this apparatus. The quantity within parentheses is a
constant for any given pair of Helmholtz coils. The value of r, the radius of the electron beam,
can be varied by changing either the accelerating potential or the Helmholtz field current. For
any given set of values, the value of e/m can be computed.
The apparatus used in this experiment is composed of a vacuum tube where an accelerated
electron beam is produced. The tube is placed between the Helmholtz coils. The magnetic field
produced by the coils bends the electron beam.
The beam of electrons in the tube is produced by an electron gun composed of a straight filament surrounded by a coaxial anode containing a single axial slit. See Figure (2.1A and 2.1B)
for schematic diagrams of the tube. Electrons emitted from the heated filament F are accelerated by the potential difference applied between F and the anode C. Some of the electrons
come out as a narrow beam through the slit S in the side of C. When electrons of sufficiently
high kinetic energy (10.4 electron volts or more) pass through the mercury vapor in the tube,
a fraction of the atoms will be ionized. On recombination of these ions with stray electrons,
the mercury spectrum is emitted with its characteristic blue color. Since recombination with
emission of light occurs very near the point where ionization took place, the path of the beam
of electrons is visible as the electrons travel through the mercury vapor.
MEASURING MASS-TO-CHARGE RATIO E/M OF THE ELECTRON
Figure 2.1: e/m apparatus.
13
14
EXPERIMENT 2
Figure (2.1A) is a sectional view of the tube and filament assembly. Figure (2.1B) is a detailed
section of the filament assembly at right angles to Figure (2.1A).
Where:
“A” Five cross bars attached to staff wire
“B” Typical path of beam of electrons
“C” Cylindrical anode
“D” Distance from filament to far side of each of the cross bars
“E” Lead wire and support for anode
“F” Filament
“G” Lead wires and supports for filament
“L” Insulating plugs
“S” Slit in cylindrical anode
The magnetic field of the Helmholtz Coils causes the stream of electrons to move in a circular
path, the radius of which decreases as the magnetic field increases. By proper control of the
magnetic field, the sharp outer edge of the beam can be made to coincide with any one of five
bars spaced at known distances from the filament.
Each coil of the pair of Helmholtz Coils has 72 turns of copper wire with a resistance of approximately one ohm. The approximate mean radius of the coil is 33 centimeters. The coils
are supported in a frame which can be adjusted with reference to the dip angle so that their
magnetic field will be parallel to the earth’s magnetic field but oppositely directed. A graduated
scale indicates the angle of tilt. A safety device prevents the coils from accidentally dropping
back to the horizontal. Wooden seats with retaining straps cradle the tube in a central plane
midway between the two coils.
2.2
Experimental Procedure
1. Place the tube in its support in the center of the frame. The end of the tube from which
the leads extend should be adjacent to that part of the frame on which connectors are
mounted. Secure the tube in place with the straps.
Orient the Helmholtz Coils so that the tube will have its long axis in a magnetic northsouth direction, as determined by a compass. Measure the magnetic inclination at this
location with a good dip needle. Tip the coils up until the plane of the coils makes an
MEASURING MASS-TO-CHARGE RATIO E/M OF THE ELECTRON
15
angle with the horizontal equal to the complement of the dip angle. The axis of the coils
should now be parallel to the earth’s magnetic field.
Figure 2.2: Circuit diagram for the e/m tube.
2. Make electrical connections to the tube as shown in Figure (2.2). Make electrical connections to the Helmholtz Coils as shown in Figure (2.3).
3. The filament of the tube has the proper electron emission when carrying 2.5 to 4.5 amperes. Exceeding 4.5 amperes will materially shorten the tube life. Good results can be
obtained using an accelerating voltage of 22.5 to 45 volts. An electron emission current of
5 to 10 mA produces a visible beam. For maximum filament life, operate the tube with
the minimum filament current necessary for proper electron emission. It is good practice
to apply about 25 volts accelerating potential to the anode before heating the filament.
Always start with a low filament current and carefully increase it until the proper electron
emission is obtained. Since the electron beam appears suddenly, it is advisable to increase
the filament current slowly to avoid overheating and possibly burning out the filament.
Apply 20 to 30 volts accelerating potential to the tube. With maximum resistance in
the filament circuit, close the switch in this circuit. Gradually decrease this resistance
until the milliammeter indicates 5 to 10 mA and the electron beam strikes the tube wall.
Rotate the tube in its cradle until the electron beam is horizontal and the bars extend
upward from the staff wire. Note that the electron beam is deflected slightly toward the
base of the tube by the earth’s magnetic field.
16
EXPERIMENT 2
Figure 2.3: Circuit diagram for Holmholtz coils.
Table 2.1: Distance of various cross bars to the filament of the e/m tube.
Crossbar Number Distance to Filament (D), meters
1
0.065
2
0.078
3
0.090
4
0.103
5
0.115
4. Once the anode voltage has stabilized send a small current through the Helmholtz Coils.
If the magnetic field of the coils straightens the beam or curves it in the opposite direction,
the coils are correctly connected. If this field increases the deflection toward the base, the
current is flowing through the coils in the wrong direction, and the connections must be
reversed. If no effect is observed on the beam, the field due to one coil is opposing that
of the other, and connections to one coil must be reversed. Make any necessary changes
and increase the field current sufficiently to cause the beam to describe a circular path
without striking the wall of the tube. The apparatus should now be ready for obtaining
the necessary data to compute the value of e/m.
In the manufacture of the tubes, the crossbars have been attached to the staff wire,
and this assembly attached to the filament assembly in precise fixtures which accurately
control the distances between the several crossbars and the filament. Distances given are
to the far side of the crossbar.
These distances are the diameters of the several circles which the electron beam will describe. Following the prescribed procedure, heat the filament until the electron beam is
visible. Allow a few minutes for warm-up and temperature equilibrium.
MEASURING MASS-TO-CHARGE RATIO E/M OF THE ELECTRON
17
Table 2.2: Data table.
Crossbar No.
D
r
V
I1
I2
I
e/m
5. Close the circuit comprising the Helmholtz Coils and adjust the value of the current until
the electron beam is straight. One method is to adjust the beam until it is aligned with a
straight edge held as close to the tube as possible. Another method is as follows. Increase
the current through the coils until the beam shows a slight curvature as compared with
the straight edge and record the value of the current. Then decrease the current until
the beam shows an equal curvature in the opposite direction and record the current. The
average of these values will be the value of the current required to straighten the beam.
This adjustment should be made carefully and accurately. When the beam is straight,
the magnetic field of the coils just equals the earth’s magnetic field. Record the value of
this current as I1 in a table similar to Table (2.2)
The value of I1 is dependent only on the accelerating voltage, V , and the earth’s magnetic
field. It is constant unless V is changed, in which case I1 must be redetermined.
6. Increase the field current until the electron beam describes a circle. Adjust its value until
the sharp outside edge of the beam strikes the outside edge of a crossbar. Record this
value as I2 in the table. The outside edge of the beam is used because it is determined
by the electrons with the greatest velocity. The electrons leaving the negative end of
the filament fall through the greatest potential difference between filament and anode,
and have the greatest velocity. It is this potential difference which the voltmeter measures.
7. Determine and record the field current required to cause the electron beam to strike each
of the other bars. Subtract I1 from each value of I2 and record as I. This is the value of the
current to be used in Equation (2.6) for computing e/m. Compute e/m for each value of I.
8. Repeat the above using a different value of accelerating voltage.
9. Calculate the average e/m ratio and it’s related error. Compare this result with the accepted value of e/m, which is 1.76 × 1011 C/kg.
18
EXPERIMENT 2
2.3
Supplementary Notes
• By subtracting the field current just necessary to overcome the earth’s field from the
total field current required to curve the beam into a given circular path, the effect of the
earth’s field on the determination of e/m is eliminated, and there is no need to know
or otherwise determine the earth’s magnetic field. However, if desired, the value of the
earth’s magnetic field may be computed using the value of I1 in Equation (2.5).
• If the magnetic field of the Helmholtz Coils is increased sufficiently to cause the electron
beam to describe a circle of small radius, then by rotating the tube slightly, the beam
will spiral either upward or downward. Reversing the filament current will reverse the
direction of the spiral (the spiraling of the beam is caused by the magnetic effect of the
filament current). A reversing switch in the filament circuit facilitates this demonstration.
• The action of this tube and Helmholtz Coils can be used in explaining the principle of a
mass spectrometer. A consideration of Equation (2.6) shows that, for charged particles
carrying the same charge but differing in mass, the radius of the circular path into which
they are caused to move by the magnetic field differs for particles of different masses.
Knowing the value of the charge and measuring the radius of the circular path, the mass
of particles can be determined. This is essentially what is done in a mass spectrometer.
Experiment 3
Michelson’s Interferometer
3.1
Introduction
An interferometer is a device that can be used to measure wavelength, length or change in
length with great accuracy by means of interference fringes. In this experiment we will use an
interferometer which was devised and built by A. A. Michelson in 1881. This interferometer
and variations of it are still in use.
The interferometer is shown schematically in Figure (3.1). Light from the source falls on mirror
M which is half-silvered i.e. it is a beam splitter. This mirror has a silver coating thick enough
to transmit half the incident light and to reflect the other half. At M the light, then, splits into
two waves. One is transmitted to mirror M1 and the other is reflected to mirror M2 . The two
mirrors M1 and M2 reflect the light waves back along their directions of incidence and the waves
will then arrive at the observation screen. The two waves are coherent since they originated
from the same source. An interference pattern will then form at the observation screen.
The two light waves travel through two different paths. If the mirrors M1 and M2 are exactly
perpendicular to each other, then the path difference between the two waves arriving at the
observation screen is simply d2 − d1 . As a result a circular fringe pattern will form on the
observation screen due to small changes in the angle of incidence of the light waves emanating
from different points on the source.
If M2 moves backward or forward, the effect is to change the path difference d2 − d1 . If M2
moves a distance λ/2 then the path difference will change by λ and a bright fringe will then
become a dark one and a dark fringe becomes a bright one. In other words, for every fringe
change from bright to dark (or from dark to bright) the moving mirror moves a distance of λ/2
of the light used. If now M2 moves a distance dm and as a result N fringes changed from bright
fringes to dark fringes, then:
λ=
2dm
N
(3.1)
20
EXPERIMENT 3
If the wavelength λ of the light used is known one then can measure distances as small as the
wavelength.
Figure 3.1: Basic components of Michelson’s interferometer.
3.2
Experimental Procedure
In this experiment we are going to measure the wavelength of a He-Ne laser using Michelson
Interferometer.
3.2.1
Setup and alignment of the Interferometer:
1. Set the interferometer on a lab table with the micrometer knob pointing toward you.
Position the laser to the left of the base.
2. Place the mirror (M1 ) that is mounted on right angle bracket in the recessed hole in the
interferometer base, and secure with a thumbscrew.
3. Place the laser on the lab table. Adjust the laser until the laser beam is approximately
parallel with the top of the interferometer base and strikes the mirror in the center. To
check that the beam is parallel with the base, place a piece of paper in the beam path,
with the edge of paper flush against the base. mark the height of the beam on the paper.
Using the piece of paper, check that the beam is at the same height at both ends of the
MICHELSON’S INTERFEROMETER
21
interferometer base.
4. Adjust the laser until the beam is reflected from the mirror right back to the laser aperture. This can be done by gently tapping the rear end of the laser transverse to the axis
of the beam.
5. Mount the beam splitter (M ), the second mirror (M2 ), and the observation screen on the
interferometer base as shown in Figure (3.1).
6. Position the beam splitter at a 45◦ angle to the laser beam so that the beam is reflected
to mirror M2 . Adjust the angle of the beam splitter as needed so that the reflected beam
hits mirror M2 near its center.
7. There should now be two sets of bright dots on the observation screen; one set comes
from the fixed mirror and the other comes from the movable mirror. Each set of dots
should include a bright dot with two or more dots of lesser brightness (due to multiple reflections). Adjust the angle of the beam splitter again until the two sets of dots
are as close together as possible, then tighten the thumbscrew to secure the beam splitter.
8. Using the knobs on the back of mirror M2 , adjust the tilt of mirror M2 until the two sets
of dots on the observation screen coincide.
9. Place the 18 mm (focal length) lens in front of the laser as shown in Figure (3.1), and
adjust its position until the diverging beam is centered on the beam splitter. You should
now see circular fringes on the observation screen. If not, carefully adjust the tilt of mirror
M2 until the fringes appear.
3.2.2
Measurements:
10. Adjust the micrometer knob to a medium reading (approximately 50 µm). In this position the relationship between the micrometer reading and the mirror M2 movement is
nearly linear.
11. Turn the micrometer one full turn counterclockwise. Continue turning counterclockwise
until the zero on the knob is aligned with the index mark. Record the micrometer reading.
When you reverse the direction in which you turn the micrometer knob, there is a small
amount of slack before the mirror begins to move. This is called mechanical backlash, and
is present in all mechanical system involving reversals in direction of motion. Beginning
22
EXPERIMENT 3
with a full counterclockwise turn, and then turning only counterclockwise when counting
fringes, eliminates errors due to backlash.
12. Adjust the position of the observation screen so that one of the marks on the millimeter
scale is aligned with one of the fringes in the interference pattern. It is easier to count
the fringes if the reference mark is one or two fringes away from the center of the pattern.
13. Rotate the micrometer knob slowly counterclockwise. Count the fringes as they pass the
reference mark. Continue until some predetermined number of fringes have passed the
reference mark (count at least 20 fringes). As you finish your count the fringes should be
in the same position with respect to the reference mark as they were when you started to
count. Read and record the final reading of the micrometer dial, the difference between
this and the original reading is the distance, dm , the amount the mirror M2 moved toward
the beam splitter. Each small division on the micrometer knob corresponds to one µm
(10−6 m) of mirror movement.
14. Record N , the number of fringe transitions counted.
15. Repeat steps 10 through 14 several times, (a minimum of five times) recording the results
each time.
16. For every set of measurement, calculate the wavelength of the laser using Equation (3.1),
then average your results.
17. Compare the measured value of the wavelength λ to the known value of 632.8 nm by
finding the percent difference.
Experiment 4
Determination of the Elementary
Charge e Using Millikan Apparatus
4.1
Introduction
The charge on an electron is a fundamental quantity of physics. This charge is the smallest
amount of charge in nature. However, the quark theory of matter, which has been gaining
strong acceptance during the past decade or so, stipulates that quarks have charges of 13 e
and − 32 e where e is the negative charge of the electron. There is experimental evidence that
particles like the neutron and the proton are composed of smaller ones. But no one has ever
observed a free quark. So far, then, the smallest charge which can be observed is the electronic
charge e.
In 1909 Robert A. Millikan reported the “oil drop experiment”. The experiment was designed
to demonstrate that electric charge is “quantized” and to measure the quantum of the electric
charge e. In the experiment small drops of oil were placed between two charged horizontal
parallel plates which subject the drops to a combination of electrical, gravitational and viscous
forces. The drops were small enough to have acquired random charges equal to small integral
multiples of the quantum of charge. Analyzing the motion of the drops under the influence of
an electric field allowed Millikan to determine the charge on an electron.
The size and mass of oil drops vary, making Millikan’s original experiment complex. In this
experiment we will use small plastic spheres of known uniform size and density to simplify the
analysis. Occasionally you might observe a fragment of a sphere or a cluster of several spheres.
However, these can be quickly recognized by their size and velocity compared to most observed
spheres, and thus can be discarded.
If the mass of a sphere under observation is m, then the gravitational force on it is given by:
Fg = mg
(4.1)
24
EXPERIMENT 4
If the electric field E is varied until the electric force FE on the sphere is upward and equals the
downward gravitational force Fg , the sphere will remain stationary. The viscous force due to
the friction between the sphere and the air is, in this case, zero because there is no net motion,
so we get:
Fg = FE
or
mg = FE
(4.2)
The electric force on the sphere is given by:
FE = qE
(4.3)
thus:
mg
(4.4)
E
The electric field between two charged parallel plates is uniform in the central region, then:
q=
E=−
dV
V
=
dr
d
Using Equation (4.4) we get:
mgd
V
where d is the distance between the plates and V is the voltage across them.
q=
(4.5)
Since m, g and d are constants then:
q∝
1
V
The mass m of the sphere may be calculated from the radius and density specified. The plate
spacing d (4.0 mm) and the acceleration due to gravity g (9.8 m/s2 ) are known. Thus to
calculate the charge q on a particular sphere, you need only measure V using a voltmeter.
When many observations are made, the resultant calculated values of q will be found to be an
integral multiple of a certain small value. This value is the fundamental unit of charge.
A sphere moving through a fluid medium at a constant velocity v is subject to a viscous
retarding force Fa , given by Stoke’s law:
Fa = Krv
(4.6)
where K is a constant which depends only on the fluid’s viscosity and r is radius of the sphere.
Since r is a constant in these experiments, Kr can be replaced by another constant C, and
Equation (4.6) becomes:
Fa = Cv
(4.7)
DETERMINATION OF THE ELEMENTARY CHARGE E USING MILLIKAN
APPARATUS
25
In the absence of an electric field, a free-falling particle quickly reaches a constant terminal
velocity due to the retarding force of the fluid. At the terminal velocity, Fg = Fa , so:
mg = Cvg
(4.8)
where vg is the free fall terminal velocity.
An electric field E acting on a sphere with charge q applies a force, FE , given by Equation
(4.3). If this force is applied upwards, it will oppose the force of gravity. When the sphere
reaches equilibrium velocity ve , the sum of the forces on the sphere must be zero. When the
field is large enough to more than overcome gravity, FE is greater than Fg ; the forces Fa and
Fg are in opposite directions as FE , so:
FE − Fa − Fg = 0
or
FE = Fa + Fg
(4.9)
Using Equations (4.3), (4.7) and (4.1) we get:
qE = Cve + mg
(4.10)
or
q=
Cve + mg
E
(4.11)
Using Equation (4.8) we get:
C
(ve + vg )
(4.12)
E
A similar equation can be derived when FE is directed downward. For a sphere of uniform size
in a constant electric field, a change in the charge q results only in a change in the equilibrium
velocity ve , i.e.:
q=
C
∆ve
(4.13)
E
where the signs of q, E, and ve are considered. If many experimental values of ∆ve are measured,
they are all found to be integral multiples of a certain small value. The same must be true for
∆q: the charge gained or lost is an exact multiple of some small charge.
∆q =
4.2
Experimental Procedure
PART I: THE CLASSICAL MILLIKAN EXPERIMENT FOR DETERMINING THE CHARGE
OF AN ELECTRON
1. Squeeze the rubber bulb several times to release some spheres into the viewing chamber.
Most spheres will be electrically charged by the friction of the injection. Move the connections in Figure (4.1) to the up (upper plate +ve and lower plate -ve) or down (upper plate
26
EXPERIMENT 4
-ve and lower plate +ve) positions and observe the effect on the spheres. Some spheres
will move up and others down, indicating that there are both negatively and positively
charged spheres.The more highly charged spheres move faster and quickly disappear from
the field of view. Vary the field intensity by means of the power supply voltage control
and comment on the effect. Keep in mind that the microscope inverts the field, and the
actual direction of motion is opposite the apparent one.
Figure 4.1: Circuit diagram.
With the connection in the center position, the plates are shorted and there is no electric
field; the spheres fall freely under gravity. Look for occasional clumps of spheres falling
more rapidly and fragments falling slower than single whole spheres. Do not use these in
your measurements.
2. Turn the connection back to the up position, so that the top plate will be positively
charged to attract negatively charged spheres. Set the applied voltage to about 200 V .
Select a single sphere that is moving slowly in the electric field and try to make it stop by
carefully adjusting the voltage. Observe it long enough to make certain it is motionless,
then record the voltage across the plates. Record the stopping voltages for at least ten
spheres, try to pick out spheres requiring the highest voltage to stop; these are the ones
with the lowest charges.
3. Using the spheres radius, density, plate separation (4.0 × 10−3 m) and stopping voltage
for each sphere, calculate and record the charge on the each sphere using Equation (4.5).
4. Graph the experimentally determined values of q versus the sphere number. The charges
on the spheres will appear to cluster around certain values. These values are integral
multiples of the charge of the electron which can then be determined.
5. Draw horizontal lines which intersect the y-axis at values of e, 2e, 3e, etc. In the conclusion
comment on the clustering of your data in relation to these lines.
DETERMINATION OF THE ELEMENTARY CHARGE E USING MILLIKAN
APPARATUS
27
PART II: TERMINAL VELOCITY AND DRIVING FORCE
Figure 4.2: Various forces acting on the oil drop.
6. To clearly show the relationship between velocity and driving force, you will need to make
three velocity measurements on each of a minimum of ten spheres (you can keep them
in your field of view by switching the electric field) under the following conditions (see
Figure (4.2)):
(a) The velocity under free fall,
(b) The velocity when the electric and gravitational fields are in opposite directions, and
(c) The velocity when the electric and gravitational fields are in the same direction.
At terminal velocity the viscous force exactly balances Fg and FE ; the “driving force” in
each part of Figure (4.2) equals the viscous force. The total force is zero.
To avoid a mixture of both positively and negatively charged particles be sure to use only
the spheres that move downward when the top plate is positive.
7. Make sure there is no electric field applied to the apparatus and inject some spheres
into the viewing chamber. Choose a slowly moving sphere in free fall (remember the
microscope inverts the field). Use a stop watch to measure the time for the sphere to
move through two graduations of the reticule (a movement of 1 mm in the chamber) in
two different parts of the field of view. The two times should essentially be the same,
indicating that the sphere has reached terminal velocity.
8. Calculate the velocity values for the sphere and record your data in a table similar to Table
(4.1). The designation “up” and “down” refer to actual direction of motion (opposite the
apparent direction).
9. For the same sphere measure its velocity in two different parts of the field of view with
an electric field applied between the plates. The two velocities should again be essentially
the same. Note its actual direction of motion. The plate voltage should be kept constant
for all measurements at about 200 V (it is not necessary to know the exact value in this
experiment).
28
EXPERIMENT 4
Table 4.1: Data Table I.
Sphere
Free Fall
Up
Down
Time (s) v (mm/s) Time (s) vup (mm/s) Time (s) vdown (mm/s)
1
2
3
4
5
6
7
8
9
10
10. Watching the same sphere reverse the electric field applied between the plates. The sphere
should reverse direction. Again measure and record the sphere’s velocity in two different
parts of the field of view, as well as its actual direction of motion.
11. To avoid using data from fragments or clusters, throw out data on any sphere with a free
fall time markedly different from the majority. If all spheres have the same mass, and the
field strength is held constant, the driving force will differ only due to the charge on the
sphere q.
12. Plot a graph of the velocity versus electric force (constant values), designating downward
force and velocity as negative and upward as positive. The graph of the measurements
on each sphere should be a straight line. That is, the three points for each sphere should
fall on a straight line. The slopes will be different (although the values for the free fall
velocity should be similar) because the charges on each sphere are different, and hence
different values of FE . The graph should show that the velocity is proportional to the
driving force.
PART III: QUANTIZATION OF CHARGE
13. As noted before, sphere clumps and fragments must not be used. Since vdown is proportional to FE + Fg and vup is proportional to FE − Fg , then the vector sum, ~vup + ~vdown for
a given sphere, is proportional to 2Fg and the difference, ~vup − ~vdown , is proportional to
2FE . We can then use the values of ~vup + ~vdown to screen out all but single spheres, and
the values of ~vup − ~vdown for the selected spheres to determine charge quanta.
14. Using the data gathered in Part II, calculate ~vup + ~vdown and ~vup − ~vdown and fill in Data
Table II. Remember that the direction of v~up and vdown
~ are opposite when adding them.
15. Examine the v~up + vdown
~ column and discard spheres for which the value is markedly
different from the average.
DETERMINATION OF THE ELEMENTARY CHARGE E USING MILLIKAN
APPARATUS
29
Table 4.2: Data Table II.
Sphere
Up
Down
Time (s) vup (mm/s) Time (s) vdown (mm/s) ~vup + ~vdown
~vup − ~vdown
1
2
3
4
5
6
7
8
9
10
16. Make a bar graph of ~vup − ~vdown for the selected spheres. The data should indicate that
these values are integral multiples of some small value and, therefore, that the charge on
the spheres are also integral multiples of some small value. See Part I for calculation of
the results.
30
EXPERIMENT 4
Experiment 5
Photoelectric Effect
5.1
Introduction
The photoelectric effect was discovered by Heinrich Hertz in 1887 during the course of experiments; the primary intent of which was confirmation of Maxwell’s theoretical prediction (1864)
of the existence of electromagnetic waves produced by oscillating electric currents.
The photoelectric effect is but one of several processes by which electrons may be removed from
the surface of a substance (we shall refer here to metal surfaces in particular; since the effects
were first observed in metals and still are most easily observed in them). These effects are:
1. Thermionic emission: Heating of a metal, which gives thermal energy to the electrons
and effectively boil them from the surface.
2. Secondary Emission: Transfer of kinetic energy from particle that strike the surface to
the electrons in the metal.
3. Field Emission: Extraction of electrons from the metal by a strong external electric field.
4. Photoelectric Emission: Described below.
The photoelectric effect occurs when electromagnetic radiation shines on a clean metal surface
and electrons are released from the surface. The valence electrons in a metal are free to move
about through its interior but are bound to the metal as a whole. It is such relatively free electrons with which we shall be concerned here. We can most simply describe the photoelectric
effect as a light beam supplying electrons with an amount of energy that equals or exceeds the
energy which binds the electrons to the surface and allowing those electrons to escape. A more
detailed description of the photoelectric effect requires a knowledge, based on experiment, of
how the several variables involved in photoelectric emission are related to one another. These
variables are the frequency ν of the light, the intensity I of the light beam, the photoelectric
current i, the electron’s kinetic energy 21 m◦ v 2 , and the chemical identity of the surface from
which the electrons emerge. The emergent electrons are called photoelectrons. The minimum
32
EXPERIMENT 5
extra kinetic energy that allows electrons to escape the material is called the work function φ.
The work function is the minimum binding energy of the electron to the material. The work
functions of alkali metals are smaller than those of other metals.
Figure 5.1: Circuit diagram for a photo cell.
Experiments carried out using an apparatus similar to the one in Figure (5.1), showed that
incident light falling on the “photocathode” ejects electrons. Some of the electrons will travel
toward the “anode” where a negative or a positive V is applied. If the voltage of the anode
is positive; the electrons will accelerated toward the anode. On the other hand, if the anode
voltage is negative; the electrons will be retarded. The stream of electrons traveling from the
photocathode to the anode form an electric current I in the circuit which is measured by the
ammeter. This current is called “photocurrent”. Those experiments established the following
facts about the photoelectric effect:
1. One can measure the kinetic energy of the photoelectrons by applying negative voltage
to the anode. The negative voltage that prevents the electrons from reaching the anode
is a measure of the kinetic energy of the photoelectrons and is called the “stopping
potential” or “stopping voltage”. The kinetic energies or the stopping potentials of the
photoelectrons are independent of the intensity of the incident light. However, the number
of electrons emitted depends on the intensity as shown in Figure (5.2).
2. The maximum kinetic energy of the emitted photoelectrons, from a given material, depends only on the frequency of the incident light. In other words, the stopping potential
depends on the frequency ν and not on the intensity as shown in Figure (5.3).
3. The smaller the work function φ of the photocathode material the smaller the minimum
frequency required to eject photoelectrons, no matter how high the intensity. Figure (5.4),
shows the minimum frequency ν◦ for different materials.
PHOTOELECTRIC EFFECT
33
Figure 5.2: Voltage vs current produced by light beams with the same frequency and different
intensities.
4. For a constant voltage applied to the anode, and a constant light frequency the photoelectron current is directly proportional to the intensity of the incident light.
5. The photoelectrons are emitted almost instantly after illuminating the photocathode independent of the intensity of the incident light.
It is possible to use a classical theory to explain the emission of electrons from metals using
electromagnetic radiation. However, such a theory could not predict all the experimental facts
mentioned above except # 4.
In 1905, Einstein was able to use the concept of light quanta introduced by Max Planck in 1900
to fully explain the photoelectric effect and was able to predict all the experimental facts of
the phenomena. For a detailed explanation of Einstein’s quantum theory of the photoelectric
effect, see any book on “Modern Physics”. We give here the important equations.
From Einstein’s quantum theory:
E = hν
(5.1)
λν = c
(5.2)
where E is the energy of the incident photon, ν is the photon’s frequency, λ is the photon’s
wavelength and h is Planck’s constant.
34
EXPERIMENT 5
hν = φ + K
(5.3)
where K is the kinetic energy of the photoelectron after emission and φ is the work function or
the energy required to take the electron away from the metal. Since K = 12 mv 2 then:
1
(5.4)
hν = φ + mv 2
2
If V◦ is the potential required to stop all photoelectrons from reaching the anode then we get:
1
eV◦ = mv 2
2
(5.5)
1 2
mv = eV◦ = hν − φ
2
(5.6)
Substitute this into (3.4) to get:
5.2
Experimental Procedure
The apparatus used in this experiment is a “photo cell” composed of a glass container under
vacuum and has a photocathode and anode. The photo cell is housed in a box which also
contains a variable voltage supply and an ammeter to measure the photocurrent. The photocurrent produced by the cell itself is extremely small (of the order of several nanoamperes).
One needs to amplify such currents to be able to detect them with reasonable accuracy. The
current from the photo cell is then introduced to an amplifier before detection. In order to
minimize the effects of any stray voltages on the measurements, the amplifier is built inside the
box and only few centimeters away from the cell’s base. The apparatus is built by “Daedalon
Corporation” and is called “EP-05”. The apparatus includes three filters (Red, Green, and
Blue) to provide spectral separation from wide spectrum light sources. In this experiment you
will use a mercury lamp as a light source.
1. Set up the EP-05 on a table such that the aperture in front of the photo cell faces the
light source.
2. Connect a voltmeter to the red and black banana jacks on the top of the EP-05 box.
They are connected across the photo cell and measure the voltage across the tube.
3. Cover the aperture with a black piece of paper, card board, or metal to set the zero
adjustment on the amplifier. Do not use your hand; it is not opaque enough. Turn on
the amplifier and adjust the “Zero Adjust” knob until the meter reads zero. It is a good
practice to check the zero adjustment frequently during the measurements to correct for
any drifts.
4. Turn the “Voltage Adjust” knob to its counter clockwise limit. The voltmeter should
read zero or very close to it.
PHOTOELECTRIC EFFECT
35
Figure 5.3: Voltage vs current produced by light beams with the same frequency and different
intensities.
5. Turn on the mercury lamp. Place the filter which transmits the shortest wavelength over
the aperture.
6. Move the EP-05 until the radiation is striking the center of the photo cell. The reading
on the output meter is helpful in making the adjustment.The radiation intensity should
be adjusted so that the meter is approximately 10 on the scale. The intensity of the
radiation at the cell can be changed by changing the distance between the EP-05 box and
the light source. The amplifier gain is quite high (1 mA on the meter is 5 × 10−9 A of
photocurrent), so if the meter goes off scale, the photo cell would not be harmed. The
meter would not be harmed either; the amplifier limits the current delivered to it.
7. Note that the output current is a function of the negative voltage applied to the anode.
As the voltage increases, fewer and fewer electrons have enough energy to reach the anode
and the current drops. The critical point is the voltage at which the current falls to zero.
8. Measure the voltage for zero current five times, resetting the zero after each measurements.
Be careful not to pass the zero current value. The current remains at zero for stopping
voltage higher than the critical value. The value you need is when the current just reaches
zero. Then calculate the average stopping voltage.
9. Place the filter which transmits the next wavelength over the aperture. Repeat steps 7
and 8 You will find that the stopping voltage of this light is greater than that for the
previous wavelength.
36
EXPERIMENT 5
Figure 5.4: Work functions of various photo cathodes.
10. Repeat step 9 until you have used all the filters.
11. Use your data and Equation (5.6) to graphically find Planck’s constant h and the work
function φ of the material of the photocathode used in the photo cell.
Experiment 6
Radioactivity
6.1
Introduction
Figure 6.1: Circuit Diagram for a Geiger Mueller Counter.
In this experiment you will study some of the characteristics of radioactive decay. You will also
learn how to measure and count the radiation produced by a radioactive source. We will use
a radioactive source which emits radiation such as gamma rays and/or charged particles like β
particles (electrons) or α particles. The radiation is detected and counted by a device called
Geiger-Mueller counter. It is not possible, however, to measure the energy of these particles
using this simple device. The Geiger-Mueller counter used in this experiment operates under
computer control. The computer will allow you to set up the counter and control the counting
procedure. The computer accumulates the data and displays the results in a graph form.
6.1.1
The Geiger-Mueller Counter
The Geiger-Mueller counter, normally known as G-M tube, is an insulated; gas filled tube which
has a conducting hollow cylinder (the cathode) concentric with a conducting wire (the anode)
as shown in Figure (6.1). The cylinder is kept at a lower potential than the central wire. When
the particles of the radiation enter the counter; they interact with the atoms of the gas and
38
EXPERIMENT 6
produce some initial ionization.
The ionization process produces ion pairs each consisting of a positive ion and an electron. If
there is no voltage across the counter, then, the electrons and positive ions will recombine to
form neutral atoms and nothing happens further. If, however, the potential difference between
the electrodes of the counter increases, the electrons will be swept away toward the central wire
and the positive ions toward the negative cylinder. As a result of the motion of the ion pairs
some charge will develop on the capacitor in the circuit. This charge will change the potential
difference across the resistor. This change is detected and amplified by the amplifier and then
counted by a counter (not shown in Figure (6.1)).
Figure 6.2: Voltage vs count rate of a G-M tube.
Figure (6.2) shows the relationship between the G-M tube voltage and the number of counts
per unit time. When the voltage is below certain value (tube threshold) the voltage is not
enough to sweep the ion pairs. Above threshold the voltage is high enough such that the ion
pairs will accelerate and produce secondary ion pairs which in turn get swept by the voltage
to the electrodes. Within a certain range of voltage above threshold, every particle enters the
tube will cause enough ionization in the tube and get detected, irrespective of its energy. The
count rate in this region is then independent of the applied voltage. This region is called the
plateau. Raising the applied voltage above this region will cause a very large amount of secondary ionization and the gas in the tube simply breaks down. The voltage region below the
plateau can be different from the one shown in Figure (6.2). The tube behavior in this region
varies with the design and the intended use of the tube.
RADIOACTIVITY
39
The G-M tube should then operate at a voltage somewhere in the plateau region. The first
part of this experiment is to reproduce the curve in Figure (6.2) and determine the operating
voltage of the G-M tube.
6.1.2
Random Nature of Radioactive Decay
The laws governing the radioactive decay process were established experimentally by Rutherford
and Soddy and states that the activity of a radioactive sample decays exponentially in time,
and in a random fashion. Let us assume that λ is the probability that a nucleus decays per
unit time. In a sample of N such nuclei, the mean number of nuclei dN decaying in a time
interval dt is :
dN = −λN dt
(6.1)
λ is called the decay constant and depends only on the particular decay process of a particular
nucleus. If N is very large, which is normally the case even in a small amount of radioactive
material, then the change in N can be considered as continuous. Integrating Equation (6.1) we
get:
N (t) = N (0)e−λt
(6.2)
Where N (0) is the number of nuclei at t = 0. The decrease in radioactivity is then controlled
by the decay constant λ. It is customary to use the inverse of λ,
τm =
1
λ
(6.3)
which is called the mean life time. The time interval τm is the time required for the sample to
decay to 1/e of its initial activity. One can then define the half lif e, T1/2 , which is the time
it takes the sample to decay to one-half its original activity, thus
T1/2
1
= e−λT1/2 = e− τm
2
(6.4)
T1/2 = τm ln(2)
(6.5)
this gives
Consider now the number of decays, n, undergone by a radioactive source in a time interval
∆t. If ∆t is short compared to the half-life of the source, then, the total activity of the source
could be considered as constant. If we measure n several times we would observe fluctuations
in its value from measurement to measurement. This is because n is related to the probability
40
EXPERIMENT 6
of decay. It can be shown that the probability of observing n decays in a time interval ∆t is
given by Poisson distribution:
P (n, ∆t) =
µn −µ
e
n!
(6.6)
where µ is the average number of counts in the interval ∆t. The standard deviation of the
Poisson distribution is:
σ=
√
µ
(6.7)
The standard deviation σ is, in general, a measure of the spread of the data around its mean
value. However, in case of Poisson distribution it is also a measure of the uncertainty associated
with our experimental attempts to determine “true” values.
6.2
6.2.1
Experimental Procedure
Determination of the Operating Voltage of the G-M Tube:
1. Turn off computer power.
2. Connect the RS-232C cable from the GMX box to the serial port of the computer.
3. Connect the G-M tube to the GMX box with the coaxial cable that is terminated with
MHV connector.
4. Turn on the power to the GMX box, then turn the computer on.
5. After the computer boots and you get the DOS prompt, type “cd windows” this will make
the windows directory the current directory.
6. Type “win” to start the windows program.
7. The program which controls the GMX box is called “gmx.exe”. The full path to this
program is “c:/windows/gmx/gmx.exe”. This means that the program resides in a folder
called “gmx” which is in a folder called “windows” which is on the hard drive called the
“c” drive.
8. To reach program “gmx.exe” you locate the icon “Main”. Use the mouse to drag the
pointer to the “Main” icon and click the mouse button to open the “Main” window.
9. Locate the icon of the “File Manager” and click on it to open its window.
RADIOACTIVITY
41
Table 6.1: Summary of the gmx program commands and the keyboard keys used to invoke
them.
Key/Keys
F1
F3
F4
F5
F6
F7
F8
Up/Dn Arrow
Plus/Minus
(Numeric Keypad)
Ctrl F2
Function
Acquire toggle
Set preset (non-acquire mode only)
Draw Grid toggle
Select Vertical Scale (Linear or Log)
Load a Binary File (non-acquire mode only)
Save a Binary File (non-acquire mode only)
Select any Linear Vertical Scale Multiplier at any
time except when in acquire mode
Vertical Scale Multiplier/Divider in steps of 2
Increment/Decrement volts in steps of 25
(non-acquire mode only). Has different meaning
in other experiments
Erase Spectrum
10. The “File Manager” window is split into two parts. Click on the image of the yellow
folder on the left half called “windows”. The contents of the “windows” folder will then
be displayed on the right half of the “File Manager” window.
11. Click on the folder named gmx. An image for folder gmx will appear below the image of
the “windows” folder on the right side. The contents of the gmx folder will be displayed
on the right side, where you will find “gmx.exe”.
12. Click on “gmx.exe” to start the program. An introductory screen appears. Press any key
to continue.
13. The Geiger Plateau Experiment screen will be loaded.
14. The program requires certain information to set the experiment. The keyboard is the
only way to interact and control the experiment. Table (6.1) lists command associated
with certain keys.
15. Press “Alt E” (the “alt” key and the “e” key simultaneously) and then the arrow keys to
select the experiment.
16. Type a vertical scale multiplier (e.g. 200) in response to “Set Vertical scale Multiplier
(10 to 500000) :” With this setting the scale can show a maximum count of (200 × 10) =
2000 without going off scale.
17. Type a preset time (in seconds) (e.g. 10) in response to “Enter preset time (10 to 600) :”.
The preset time is the time the program use to accumulate data before stopping to take
42
EXPERIMENT 6
another measurement. Since the uncertainty in counting experiments is the square root
of the count, one then should select a time long enough to accumulate certain number of
counts which has an error of about 5% or less.
18. Type an initial tube voltage of 250V in response to “Select initial voltage (10 to 1200) :”.
19. Place Co60 source under the G-M tube and press F 1 to start acquire. Data will be
acquired at 25 volt interval until the maximum voltage of 1200 is reached or F 1 key is
pressed.
20. During acquire, counts are plotted inside the data window. If counts are out of range for
the given scale, dots are plotted near the upper boundary of the data window. You can
always press Up/Down arrow keys to adjust the vertical scale. You can also press F 4 to
draw/erase grids on the screen.
21. You can save your data on a disk as a binary file or as an ASCII file, for later reference or
for printing on another computer with a printer. Press “Alt F” and select “Save a Binary
File” or “ASCII File Save” and enter the file name. There is no need for an extension to
the file name. The extension is added automatically to the file name depending on the
type of the file. If you save a file under the name “geiger” then its actual name will be
“geiger.spm” for binary files and “geiger.asc” for ASCII files.
22. From the plot of counts vs tube voltage choose an operating voltage which close to lower
end of the plateau region. Record this value in your lab report and use it in all subsequent
experiments.
6.2.2
Statistics of Counting Experiments (Random Nature of Radioactive Decay)
1. Clear the data of the previous experiment using Ctrl F2.
2. Press “Alt E” and then the arrow keys to select the experiment.
3. Type a preset time (in seconds) (e.g. 5) in response to “Enter preset time (10 to 600) :”.
See step 17 in the previous experiment.
4. Type an initial tube voltage in response to “Select initial voltage (10 to 1200) :”. Use the
operating voltage determined in the plateau experiment.
5. Enter total number of readings; at least 50 readings should be taken in this experiment.
6. This experiment does not use any vertical or horizontal scales. So, Up/down keys will
have no effect.
7. Place the Co60 in the lowest ’shelf’ under the G-M tube and press F 1 to start.
RADIOACTIVITY
43
8. Individual counts are displayed after each measurement. At the end of the acquire process
you will have a list of the all the measurements as well as (avg - n), (avg - n)2 and the
standard deviation σ.
9. Divide your data into some 10 or 15 ranges. Plot the frequency of the data (the number
of data point within a range) vs middle value of each range.
10. Use the average of all the data and the middle value of each range to calculate the Poisson
distribution using Equation (6.6) and the standard deviation using Equation (6.7). Plot
the calculated Poisson distribution to denote the standard deviation on the same graph
as the data. Does the data fit the Poisson distribution?
44
EXPERIMENT 6
Experiment 7
Radioactivity II
7.1
Introduction
In this part of the radioactivity experiment we will investigate the effect of distance upon the
intensity of radiation. We will also investigate the absorption of radiation in matter.
7.1.1
Inverse Square Law
Radiation is emitted from the source in all directions. However, only a small fraction of the
emitted radiation actually enters the detector, the G-M tube. This fraction is a function of the
distance d from the source (assumed to be point source) to the G-M tube, see Figure (7.1).
If the source is moved further away from the tube, the amount of radiation entering the tube
should decrease with the square of the distance if the inverse square law applies. As an α or a
β source is moved away from the tube, the activity will decrease more rapidly than predicted
by the inverse square law as these particles are easily absorbed in air.
7.1.2
Absorption of Radiation
The absorption of radiation in matter is governed by an exponential decay law. To demonstrate
this, assume that radiation of intensity I0 is incident on a slab of thickness x. The change in
intensity as the radiation passes through the slab is proportional to the thickness and to the
incident intensity, see Figure (7.2).
∆I = −µI0 x
(7.1)
where the constant of proportionally µ is called the “absorption coefficient”. If the energy of
the incident radiation is constant, then µ is independent of x and the integration of Equation
(7.1) yields:
46
EXPERIMENT 7
Figure 7.1: Geometry of a point source and a G-M tube.
I = I◦ e−µx
(7.2)
where I and I◦ are the intensity of the radiation after passing through a thickness x of the
material, and the initial intensity of the radiation, respectively.
7.2
7.2.1
Experimental Procedure
Inverse Square Law:
1. Determine the operating voltage of the G-M tube as outlined in Experiment 6.
2. Press “Alt E” and then the arrow keys to select the experiment.
3. Type a vertical scale multiplier (e.g. 200) in response to “Set Vertical scale Multiplier
(10 to 500000) :” With this setting the scale can show a maximum count of (200 × 10) =
2000 without going off scale.
4. Type a preset time (in seconds) (e.g. 10) in response to “Enter preset time (10 to 600) :”.
The preset time is the time the program use to accumulate data before stopping to take
another measurement. Since the uncertainty in counting experiments is the square root
of the count, one then should select a time long enough to accumulate certain number of
counts which has an error of about 5% or less.
RADIOACTIVITY II
47
Figure 7.2: Absorption of radiation in a slab.
5. Select a tube voltage that is on the plateau of the G-M tube.
6. Initial distance selected by default is “distance = (distance × 1)”. This is displayed near
the right end corner of the bottom of the screen. You can select different distances by
pressing “PLUS/MINUS” keys on the numeric keypad.
7. Press F 1 to start. Total count is plotted at the end of each preset interval. Counting stops
at the end of each preset interval to allow time to place the source at a different distance.
To continue counting with the same distance, press F 1 again. Counts are plotted in the
same location. You may have a minimum of 10 readings for each distance. Move the
source to the next position and press F 1 again to start acquiring data.
8. Repeat step 7 until all desired positions have been counted and save your data.
9. Plot a graph of the number of counts vs.
1
.
d2
10. In the conclusion explain whether the inverse square law applies.
7.2.2
Linear Absorption coefficient:
11. Select the experiment.
12. Enter the necessary parameters to set up the experiment.
48
EXPERIMENT 7
13. Place the absorber in between the G-M tube and the source.
14. Press F 1 to start. Total count is plotted at the end of each preset interval. Counting
stops at the end of each preset interval to allow time to change the absorber. To continue
counting with the same absorber, press F 1 again. Counts are plotted in the same location.
You may have a minimum of 10 readings for each absorber. Change the absorber and
press F 1 again to start acquiring data.
15. Repeat step 14. until all desired absorbers have been tested and save your data.
16. Tabulate the absorber thicknesses in cm by dividing the given mg/cm2 for each slab by
the known density of the metal (ρAl =2700 mg/cm3 and ρP b =11360 mg/cm3 ).
17. Graph lnI vs. x (convert x values to m) for the results of each of the materials.
18. Find the absorption coefficients for the different materials from their respective graphs.
Appendix A
Error Analysis
A.1
Measurements and Experimental Errors
The word “error” normally means “mistake”. In physical measurements, however, error means
“uncertainty”. We measure quantities like temperature, electric current, distance, time, speed,
etc ... using measuring devices like thermometers, ammeters, tape measures, watches etc.
When a measurement of a certain quantity is performed, it is important to know how accurate
or precise this measurement is. That is, a quantitative evaluation of how close we think this
measurement is to the “true” value of the quantity must be established. An experimenter must
learn how to assess the uncertainties or errors associated with a physical measurement so that
he/she can convey a true picture of just how accurately a certain quantity has been measured.
In general there are three possible sources or types of experimental errors: systematic, instrumental, and statistical. The following are guidelines which are commonly followed in defining
and estimating these different forms of experimental errors.
A.1.1
Systematic Errors:
These errors occur repeatedly (systematically) every time a measurement is made and in general
would be present to the same extent in each measurement. They arise because of miscalibration
of a measuring device, ignoring some physical assumptions that should be taken into account
when using an instrument, or simply misreading a measuring device. An example of a systematic error due to miscalibration is the recording of time at which an event occurs with a
watch that is running late; every event will then be recorded late. Another example is reading
an electric current with a meter that is not zeroed properly. If the needle of the meter points
below zero when there is no current, then the reading of any current will be systematically less
than the true current.
As an example of a systematic error due to ignoring some physical assumptions in a measurement, consider an experiment to measure the speed of sound in air by measuring the time it
takes sound (from a gun shot, for example) to travel a known distance. If there is a wind
50
APPENDIX A
blowing in the same direction as the direction of travel of the sound then the measured speed
of sound would be larger than the speed of sound in still air, no matter how many times we
repeated the measurement. Clearly this systematic error is not due to any miscalibration or
misreading of the measuring devices, but due to ignoring some physical factors that could influence the measurement.
An example of a systematic error due to misreading an instrument is parallax error. Parallax
error is an optical error that occurs when a laboratory meter or similar instrument is read from
one side rather than straight on. The needle of the meter would then appear to point to a
reading on the side opposite to the side the meter is being looked at.
There is no general rule for the estimation of systematic errors. Each experimental situation
has to be investigated individually for the possible sources and magnitudes of systematic errors.
Care should be taken to reduce as much as possible the occurrence of systematic errors in a
measurement. This could be done by making sure that all instruments are calibrated and
read properly. The experimenter should be aware of all physical factors that could influence
his/her measurement and attempt to compensate for or at least estimate the magnitude of such
influences. A widely used technique for the elimination of systematic errors is to repeat the
measurement in such a way that the systematic error adds to the measured quantity in one
case and subtracts from it in another. In the example about measuring the speed of sound
given above, if the measurement was repeated with the position of the source and detector of
the sound interchanged, the wind speed would diminish the sound speed in this case, instead
of adding to it. The effect of the wind speed will thus cancel when the average of the two
measurements is taken.
A.1.2
Instrumental Uncertainties:
There is a limit to the precision with which a physical quantity can be measured with a given
instrument. The precision of the instrument (which we call here the instrumental uncertainty)
will depend on the physical principles on which the instrument works and how well the instrument is designed and built. Unless there is reason to believe otherwise, the precision of an
instrument is taken to be the smallest readable scale division of the instrument. In this course
we will report the instrumental uncertainty as one half of the smallest readable scale division.
Thus if the smallest division on a meter stick is 1 mm, we report the length of an object whose
edge falls between the 250 mm and 251 mm marks as 250.5±0.5 mm. If the edge of the object
falls much closer to the 250 mm mark than to the 251 mm mark we may report its length as
250.0±0.5 mm. Even though we might be able to estimate the reading of a meter to, let’s say,
1
of the smallest division we will still use one half of the smallest division as our instrumental
10
uncertainty. Our assumption will be that had the manufacturer of the instrument thought that
1
his/her device had a precision of 10
of the smallest division, then he/she would have told us so
explicitly or else would have subdivided the reading into more divisions. For meters that give
out a digital reading, we will take the instrumental uncertainty again to be one half of the least
ERROR ANALYSIS
51
significant digit that the instrument reads. Thus if a digital meter reads 2.54 mA, we will take
the instrumental error to be ±0.005 mA, unless the specification sheet of the instrument says
otherwise.
A.1.3
Statistical Errors:
Several measurements of the same physical quantity may give rise to a number of different
results. This random or statistical difference between results arises from the many small and
unpredictable disturbances that can influence a measurement. For example, suppose we perform an experiment to measure the time it takes a coin to fall from a height of two meters
to the floor. If the instrumental uncertainty in the stop watch we use to measure the time is
1
second, repetitive measurement of the fall time will reveal different results.
very small, say 100
There are many unpredictable factors that influence the time measurement. For example,
the reaction time of the person starting and stopping the watch might be slightly different
on different trials; depending on how the coin is released, it might undergo a different number of flips as it falls each different time; the movement of air in the room could be different
for the different tries, leading to a different air resistance. There could be many more such
small influences on the measured fall time. These influences are random and hard to predict;
sometimes they increase the measured fall time and sometimes they decrease it. Clearly the
best estimate of the “true” fall time is obtained by taking a large number of measurements of
the time, say t1 , t2 , t3 , ...tN , and then calculating the average or arithmetic mean of these values.
The arithmetic mean, t is defined as:
t=
N
(t1 + t2 + t3 + ... + tN )
1X
ti
=
N
N i=0
(A.1)
where N is the total number of measurements.
To indicate the uncertainty or the spread in these measured values about the mean, a quantity called the standard deviation is generally computed. The standard deviation of a set of
measurements is defined as:
s
σsd =
1
[(t1 − t)2 + (t2 − t)2 + (t3 − t)2 + ... + (tN − t)2 ]
N
or
σsd =
v
u
u
t
N
1 X
∆t2
N i=1 i
(A.2)
where ∆ti = (ti − t)) is the deviation and N is the number of measurements. The standard
deviation is a measure of the spread in the measurements ti . It is a good estimate of the error
or uncertainty in each measurement ti . The mean, however, is expected to have a smaller
52
APPENDIX A
uncertainty than each individual measurement. We will show in the next section that the
uncertainty in the mean is given by:
σsd
δt = √
N
(A.3)
According to the above equation, if four measurements are made, the uncertainty in the mean
will be two times smaller than that of each individual measurement; if sixteen measurements are
made the uncertainty in the mean will be four times smaller than that of an individual measurement and so on; the large the number of measurements, the smaller the uncertainty in the mean.
Most scientific calculators nowadays have functions that calculate averages and standard deviations. Please take the time to learn how to use these statistical functions on your calculator.
Significance of the standard deviation: If a large number of measurements of an observable are made, and the average and standard deviation calculated, then 68% of the measurements will fall between the average minus the standard deviation and the average plus the
standard deviation. For example, suppose the time it takes a coin to fall from a height of
two meters was measured one hundred times and the average found to be 0.6389 s and the
standard deviation 0.10 s. Then approximately 68% of those measurements will be between
0.5389 s (0.6389-0.10) and 0.7389 s (0.6389+0.10), and of the remaining 32 measurements approximately 16 will be larger than 0.7389 s and 16 will be less than 0.5389 s. In this example
).
the statistical uncertainty in the mean will be 0.010 s ( √0.10
100
A comment about significant figures: The least significant figure used in reporting a result
should be at the same decimal place as the uncertainty. Also, it is sufficient in most cases to
report the uncertainty itself to one significant figure. Hence in the example given above, the
mean should be reported as 0.64±0.01 s.
A.2
Combining Statistical and Instrumental Errors
In the following discussion we will assume that care has been taken to eliminate all gross systematic errors and that the remaining systematic errors are much smaller than the instrumental
or statistical ones. Let IE stand for the instrumental error in a measurement and SE for the
statistical error, then the total error, TE, is given by:
TE =
q
(IE)2 + (SE)2
(A.4)
In a good number of cases, either the IE or the SE will be dominant and hence the total error
will be approximately equal to the dominant error. If either error is less than one half of the
other then it can be ignored. For example, if, in a length measurement, SE = 1 mm and IE =
0.5 mm, then we can ignore the IE and say TE = SE = 1 mm. If we had done the exact calculation (i.e. calculated TE from the above expression) we would have obtained TE = 1.1 mm.
ERROR ANALYSIS
53
Since it is sufficient to report the error to one significant figure, we see that the approximation
we made in ignoring the IE is good enough.
Equation (3) suggests that the uncertainty in the mean gets smaller as the number of measurements is made larger. Can we make the uncertainty in the mean as small as we want by
simply repeating the measurement the requisite number of times? The answer is no; repeating
a measurement reduces the statistical error, not the total error. Once the statistical error is
less than the instrumental error, repeating the measurement does not buy us anything, since
at that stage the total error becomes dominated by the IE. In fact, you will encounter many
situations in the lab where it will not be necessary to repeat the measurement at all because
the IE error is the dominant one. These cases will usually be obvious. If you are not sure then
repeat the measurement three or four times. If you get the same result every time then the IE
is the dominant error and there is no need to repeat the measurement further.
A.3
Propagation of Errors
Very often in experimental physics a quantity gets “measured” indirectly by computing its
value from other directly measured quantities. For example, the density of a given material
might be determined by measuring the mass m and volume V of a specimen of the material
. If the uncertainty in m is δm and the uncertainty
and then calculating the density as ρ = m
V
is V is δV , what is the uncertainty in the density δρ? We will not give a proof here, but the
answer is given by:
δρ =
s

2
δm 



ρ 
m


+
2
δV 


V
More generally, if N independent quantities x1 , x2 , x3 , ... are measured and their errors δx1 , δx2 , δx3 ,
... determined, and if we wish to compute a function of these quantities
y = f (x1 , x2 , x3 , ..., xN )
then the error on y is:
s
σy =
v
u
N
uX
∂y
∂y
∂y
∂y
[
δx1 ]2 + [
δx2 ]2 + [
δx3 ]2 + ... = t [
δxi ]2
∂x1
∂x2
∂x3
∂x
i
i=1
(A.5)
∂y
Here ∂x
is the partial derivative of f with respect to xi . Note that the x0i s do not have to be
i
measurements of the same observable; x1 could be a length, x2 a time, x3 a mass and so forth.
It makes the notation easier to call them x0i s rather than L,t, m and so forth.
Applying the above formula to the most common functions encountered in the lab, the following
equations can be used in error analysis.
54
A.3.1
APPENDIX A
Linear functions:
y = Ax
(A.6)
where A is a constant (i.e. has no error associated with it), then
δy = Aδx
A.3.2
(A.7)
Addition and subtraction:
z =x±y
(A.8)
then
δz =
q
δx2 + δy 2
(A.9)
Clearly if
z = Ax ± By
(A.10)
q
(A.11)
where A and B are constants, then
δz =
A.3.3
(Aδx)2 + (Bδy)2
Multiplication and division:
z = xy
or
z=
x
y
(A.12)
then we can show that:
δz
=
z
v
u 2
u δx
t
 

x
 2
δy

 
+
y
(A.13)
Note: This method should not be used in cases including exponents.
A.3.4
Exponents:
z = xn
(A.14)
where n is a constant, then we can similarly show that:
nδx
δz
=
z
x
(A.15)
ERROR ANALYSIS
55
For the more general case where
z = Axn y m
(A.16)
where A, n, m are constants, the error would be given by:
v
u


2
2
δz u
nδx 
mδy 



 +


= t
z
x
y
(A.17)
Examples on the application of the above formula follow:
r 
2
 δx 
z = Axy
δz
z
=
z = Ax2
δz
z

= 2 δx
x
√
z=A x
δz
z

= 12  δx
x
δz
z
=
z=
A
x
A.3.5
x





2

+  δy
y
δx
x
Other frequently used functions:
y = Asin(Bx)
δy = ABcos(Bx)δx
y = ex
δy = ex δx
y = ln x
δy = x1 δx
Often, the uncertainty in a quantity y is not reported as the absolute uncertainty δy but rather
as a percentage uncertainty, 100%× δy
. For example, if the length of an object is measured to be
y
0.3
25.6 mm, with an uncertainty of 0.3 mm, then the percentage uncertainty is 1% (100% × 25.6
).
A.4
Percent Difference:
Calculating percent difference when comparing results is often useful. The appropriate formula
is:
P ercent Dif f erence =
|T heoretical V alue − Experimental Result|
· 100%
|T heoretical V alue|
(A.18)