Name: Letter: PHYS 2419 FALL 2011 COURSE WEBSITE: http://people.virginia.edu/~ecd3m/2419/Fall2011/ LABORATORY MANUAL DEPARTMENT OF PHYSICS UNIVERSITY OF VIRGINIA COURSE INSTRUCTOR: E. CRAIG DUKES EMAIL: [email protected]LABORATORY SUPERVISOR: LARRY SUDDARTH EMAIL: [email protected]
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Name: Letter: PHYS 2419 - University of Virginiapeople.virginia.edu/~ecd3m/2419/Fall2011/manual/manual.pdfPHYSICS 2419 WORKSHOP MANUAL Contents Page Introduction 1 Lab 1 Electrostatics
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It is, however, possible to produce linearly polarized
electromagnetic waves. In other words, waves whose electric
field vector only oscillates in one direction. Look again at
Figure 2. It schematically shows a linearly polarized
electromagnetic wave polarized in the x-direction.
The electric field of a plane wave of wavelength λ, propagating in
the z-direction and polarized in the x-direction, can be described
by:
−= )(
2sin ctzExx
λ
πiE , (2)
where Ex is the vector of the electric field, Ex its amplitude, and i
the unit vector in the x-direction. A wave of the same wavelength,
polarized in the y -direction, is described by:
+−= φ
λ
π)(
2sin ctzEyy jE . (3)
Here, j is the unit vector in the y-direction and φ is a constant that
accounts for the possibility that the two waves might not have the
same phase. From two such waves, one can construct all plane
waves of wavelength λ traveling in the z-direction.
L11-4 Lab 11 - Polarization
University of Virginia Physics Department PHYS 2419, Fall 2011
y
x
E
a) b) c)
E E
x
y y
x θ
Figure 4
If both x- and y-components are present and their phase difference
is zero (or 180°), the wave will be linearly polarized in a direction
somewhere between the x -direction and the y -direction, depending
on the relative magnitudes of Ex and Ey (see Figure 4a).
Mathematically such a wave is described by:
−±=+= )(
2sin)( ctzEE yxyx
λ
πjiEEE , (4)
where the plus sign refers to a phase difference of zero and the
minus sign to one of 180° (π radians). The angle θ between this
polarization direction and the x -direction is given by
x
y
E
E=θtan . (5)
If the phase shift is not zero (or 180°), the wave will not be linearly
polarized. While we will only be investigating linear polarization
in this lab, it is useful to know something about other types of
polarization. Consider the case where the magnitudes are equal,
but the phase shift is ±90° (± π/2 radians). In other words:
yx EE = and 2
πφ = ± , (6)
The resulting wave, called a circularly polarized wave, can be
written:
2 2
sin ( ) cos ( )x y E z ct z ctπ π
λ λ
= + = − ± −
E E E i j (7)
by making use of the fact that απα cos)2/(sin ±=+ . With the
plus sign, this equation describes a wave whose electric field vector,
E, rotates clockwise in the x -y plane if the wave is coming toward
the observer. Such a wave, illustrated by Figure 4b, is called a right
circularly polarized wave. With the minus sign, the equation
describes a left circularly polarized wave.
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University of Virginia Physics Department PHYS 2419, Fall 2011
With the phase shift still ±90°, but with different magnitudes
2
EE yx
πφ ±=≠ and , (8)
the E vector will still rotate clockwise or counterclockwise but will
trace out an ellipse as shown Figure 4c.
With thermal sources, there is a random mix of different Ex, Ey,
and φ values. The resulting wave will be unpolarized.
Polarized electromagnetic waves can be obtained in two ways:
1. by using sources, such as certain lasers, that produce only waves
with one plane of polarization, or
2. by polarizing unpolarized waves by passing them through a
polarizer, a device that will let only waves of one particular
plane of polarization pass through.
Some sources of electromagnetic waves generate linearly polarized
waves. Examples include the microwave generator we'll use today
as well as some types of lasers. Other sources generate unpolarized
waves. Examples include thermal sources such as the sun and
incandescent lamps.
One way of producing linearly polarized electromagnetic waves
from unpolarized sources is to make use of a process that directs
waves of a given polarization in a different direction than waves
polarized in the perpendicular direction. Earlier we noted that the
electric field of an electromagnetic wave incident upon a wire
induces an oscillating current in the wire. Some energy will be lost
through resistive heating, but most will be re-radiated (scattered).
Only the component of the oscillating electric field that is parallel
to the wire will induce a current and be scattered. The electric
field component perpendicular to the wires, on the other hand, will
be essentially unaffected by the wires (assuming a negligible wire
diameter). Hence, both the scattered and unscattered
electromagnetic waves are linearly polarized.
For microwaves, we can (and will) use an array of actual wires.
For visible light, we use a Polaroid filter. Polaroid filters are
made by absorbing iodine (a conductive material) into stretched
sheets of polyvinyl alcohol (a plastic material), creating, in effect,
an oriented assembly of microscopic “wires”. In a Polaroid filter
the component polarized parallel to the direction of stretching is
absorbed over 100 times more strongly than the perpendicular
component. The light emerging from such a filter is better than
99% linearly polarized.
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University of Virginia Physics Department PHYS 2419, Fall 2011
Polarizer P
θ
Ei
Ee
p
Figure 5
A polarizer will only pass the components of an electromagnetic
wave that are parallel to its polarizing axis. Figure 5 shows
polarized electromagnetic waves incident on a polarizing filter, P
(shown as a wire array).
The electric field of the incident wave (Ei) is oriented at an angle θ
relative to the polarization axes of P. Let p be a unit vector along
the polarization axis of the polarizer. The effect of the polarizer,
then, is to “project out” the component of Ei that is along p:
Ee = p (p·Ei) = Ei cos θ p. Because the intensity of an
electromagnetic wave is proportional to the square of its electric
field amplitude, it follows that the intensity of the electromagnetic
waves exiting the analyzer is given by:
2cose i
I I θ= . (9)
This is known as Malus’ Law, after the French physicist who
discovered the polarizability of light.
Initially unpolarized electromagnetic waves can be thought of as a
mixture of all possible polarizations. Each possible polarization
will be attenuated according to Malus’ law, and so the total
intensity will be the initial intensity times the average of 2cos θ
(which is 1/2). In other words, the intensity is reduced to one half
of the incident intensity.
Except in the case where θ is zero (or 180°), Ee (the electric field
of the electromagnetic waves exiting the polarizer) will have a
component that is perpendicular to Ei. If we place yet another
polarizer after P (call it P') with its polarization axis right angles to
incident wave’s polarization axis, we will get electromagnetic
waves out whose polarization is orthogonal to the incident waves’
polarization. We have effectively rotated the polarization of the
incident waves (with some loss of intensity). Applying Malus’
Law, we get:
2 2cos cosi
I I θ θ′ ′= (10)
where θ is the angle between the initial polarization and the first
polarizer, P, and θ' is the angle between P and the second polarizer,
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University of Virginia Physics Department PHYS 2419, Fall 2011
P'. But P' is at right angles to the initial wave's polarization, so
90θ θ ′+ = ° . Hence, 2 2cos sinθ θ′ = . Using another trigonometric
identity (sin 2θ = 2 sin θ cosθ), we finally get 20.25 sin 2i
I I θ′ = .
We can see we get the maximum transmission when θ = 45°
(sin 2×45° = 1) and that it is one quarter of the intensity of the incident polarized waves (Ii).
INVESTIGATION 1: MICROWAVE POLARIZATION
For this experiment, you will need the following:
• Gunn diode microwave transmitter
• Microwave receiver
• Wire grid polarizer
CAUTION: DO NOT ALLOW THE RECEIVER’S METER
TO PEG AT ANY TIME!
To peg the meter means to allow the needle to go beyond the
maximum value on the scale. If you find the meter pegged, immediately turn down the sensitivity and/or move the receiver
away from the microwave generator!
Activity 1-1: Polarization of Microwaves from a Gunn Diode
Inside the microwave generator is a solid state device called a Gunn diode. When a DC voltage is applied to a Gunn diode,
current flows through it in bursts at regular intervals. For your diode, these bursts come at 9.52 × 10
-11 seconds apart causing, in
addition to the dc current, an ac current at 1.05 × 1010
Hz (10.5 GHz). As a result, a large AC voltage, oscillating at that
frequency, is present across the slot, and so a wave is radiated from the horn. The electric field of this wave oscillates in the same
orientation as the Gunn diode. The polarization of an electromagnetic wave is determined by the direction of the electric vector E. The magnetic field B encircles the current in the Gunn
diode and so emanates in the orientation perpendicular to E.
Important Note: The Gunn diode is place inside the generator in a
way that the electric field will oscillate vertically when the knob on
the back is placed at 0º.
Just inside the horn of the receiver is a microwave detector. In
addition, there is some circuitry, which amplifies the signals received by the detector and outputs this amplified signal to a
d’Arsonval meter and to an external output. The sensitivity (labeled METER MULTIPLIER) is controlled via two knobs. The
VARIABLE SENSITIVITY knob allows for fine adjustment. As
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you turn up the sensitivity (from 30 to 1), the signal is amplified more and more.
Generator
Receiver
Figure 6
1. Set up the generator and receiver as shown in Figure 6, with about 75 cm between the faces of the horns.
Prediction 1-1: With what relative orientation of the transmitter and receiver do you expect to find minimum intensity? What does
this tell you about the electromagnetic microwaves?
Set the knobs on back of both pieces so the angle indicator is at 0°. Adjust the sensitivity on the receiver to obtain a signal near 0.5 on
the meter. If you cannot achieve this with a sensitivity of 10 or 3, move the receiver closer to the generator. Rotate the receiver and
verify that it is sensitive to the polarization of the wave. Return the receiver angle to 0º.
Question 1-1: Does it make sense that maximum intensity is obtained when both generator and receiver are oriented the same
way? Explain why. Why does the received signal go to zero when they are at 90º with respect to one another?
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University of Virginia Physics Department PHYS 2419, Fall 2011
Activity 1-2: Wire Grid Polarizer
Prediction 1-2: With the generator and receiver oriented the same
way, what orientation (relative to the generator) of a wire grid placed in between them will give the maximum received intensity?
Prediction 1-3: With the generator and receiver oriented at 90°
with respect to one another, what orientation (relative to the generator) of a wire grid placed in between them will give the
maximum received intensity?
1. Make sure that the generator and the receiver are oriented the same way: with the E field horizontal (indicators at ±90°).
2. Insert the wire grid polarizer between the generator and the receiver so that the wires are initially oriented horizontally
(parallel to the direction of the E field). Slowly rotate the polarizer so that the wires become perpendicular to the E field.
Question 1-2: With what relative orientation(s) of the polarizer did the receiver indicate the highest intensity? The lowest?
3. Rotate the receiver’s angle by 90º so that the generator and
receiver are orthogonal and turn up the sensitivity to 1.
4. Repeat step 2.
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University of Virginia Physics Department PHYS 2419, Fall 2011
Question 1-3: With what orientation(s) of the polarizer did the receiver indicate the highest intensity? The lowest?
NOTE: Turn off your receiver and unplug the generator.
INVESTIGATION 2: POLARIZATION OF A HIGH-INTENSITY LAMP
NOTE: IN THE REMAINDER OF THE WORKSHOP WE
WILL INVESTIGATE THE POLARIZATION OF VISIBLE
LIGHT. FOR THE NEXT THREE INVESTIGATIONS, IT
WILL BE NECESSARY TO TURN OFF ALL OF THE
LIGHTS IN THE LAB TO OBTAIN THE BEST RESULTS.
In this investigation, the unpolarized light from a high-intensity lamp will be linearly polarized. This polarization will be
investigated with a second Polaroid analyzer. In addition, a third polarizer will be added to investigate the effect of the orientation
of a third polarizer on the intensity.
For this you will need the following:
• Optical bench with lens holders
• Polarizers
• Polarized light demonstrator kit
• Goniometer
• Small support stand
• Desk lamp (high intensity light source)
• Light sensor and cable
Activity 2-1: Linearly Polarized Light and Malus’ Law
Note: The light sensor that will be used for the rest of the experiments is a photodiode with a sensitivity that ranges from 320 nm to 1,100 nm. Make sure not to allow the output voltage
from the sensor to go above 4.75 volts. At this point, the sensor
is saturated and you will not get accurate readings.
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University of Virginia Physics Department PHYS 2419, Fall 2011
Figure 7
1. Set up the lamp, polarizers, and light sensor as shown in Figure 7. DO NOT TURN YOUR LAMP ON YET. Make
sure your lamp is on the opposite end of the table from the computer and is pointing towards the wall, not towards the
center of the room. We want to minimize the interference of the light coming from the desk lamp into each other’s light
sensor.
2. The heat-absorbing filter (item #1 in the box of components)
should be mounted on the small support stand in between the light source and the first polarizer. Place it as close to the
polarizer as you can so that little, if any, light can get into the polarizer without first passing through the heat filter.
3. Ensure that the heights of the light, heat absorbing filter, polarizers and light sensor are lined up. Your lamp can now be
turned on.
4. Look through the analyzer at the lamp. Play around with the
relative orientation of the two polarizers. Record your observations.
ALWAYS PLACE THE HEAT ABSORBING FILTER
BETWEEN THE LIGHT AND THE FIRST POLAROID
FILTER TO BLOCK THE INFRARED LIGHT AND
PREVENT HEAT DAMAGE
Note: The infrared light emitted from the lamp will not be
polarized by the filters, but will be seen by the photodetector.
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University of Virginia Physics Department PHYS 2419, Fall 2011
5. Connect the light sensor to channel A in the PASCO interface.
6. Open the experiment file named L11A2-1 Linearly Polarized.
There should be a data table when you open the file.
7. In the data table, the first column will be the values for angle
that you enter. The units for intensity are not volts but Lumens. However, the output of the light sensor probe in volts
is directly proportional to the light intensity. Never let the
output from the light sensor exceed 4.75 V
8. Ensure that the light from the lamp is incident on the heat absorbing filter, then travels through the first polarizer, then
through the analyzer, and then onto the light sensor.
9. Make sure that the two polarizers are aligned (both at 0º).
10. Press Start to begin collecting data. The output from the light sensor will be shown in the digits window and in the third
column of the first row. A typical voltage reading when the lamp and sensor are 50-60 cm apart is in the range of 0.5 to
1 V. There is a sensitivity switch on the light sensor that you may need to adjust.
11. Set the first polarizer to 0º and the analyzer to –90º (counterclockwise value; see Figure 8 for one possible
convention). The Polaroid Filters we are using allow the
electric field E vector of the transmitted light to oscillate in the
direction of the indicator tab on the Polaroid.
12. When you feel that the
reading has stabilized, press Keep. A box will pop up that
asks you the angle of the polarizer. Type in “-90” and
press Enter.
13. Adjust the analyzer to an
angle of -80º. The voltage output will now be shown as
before. Press Keep and type in the angle.
14. Adjust the angle of the analyzer in 10º steps from
-90º to 90º. Repeat step 12 until all of the values are entered, putting in the respective
values for the angles. This can go rather quickly with one person changing the angle and another person operating the
computer. Once Keep has been selected, the next angle can be changed by one group member while another is entering the
angle into the computer.
Figure 8
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University of Virginia Physics Department PHYS 2419, Fall 2011
15. When you are finished entering data, click on the red square next to Keep to stop data collection.
16. Print out your table for your report. Only print one per group.
17. At the bottom of the screen, there should two graphs
minimized. Bring up the graph titled I vs. Angle so you can see the graph of your light intensity plotted versus angle. If
you see a fit to your data, you have brought up the wrong graph.
Question 2-1: What does your graph look like? Does it follow the curve you would expect?
18. Minimize this graph, and maximize the second graph entitled
Fit Malus. You will see your data plotted along with a fit. You could have easily entered this fit into Data Studio
yourself, but we have done it for you to save time. We have fit straight line (y = mx + b) to I versus cos
2θ.
19. Record the fit parameters m and b:
m b
Question 2-2: Discuss the physical meanings of m and b.
Question 2-3: Is it possible that b is not constant? Explain. How could you minimize b?
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University of Virginia Physics Department PHYS 2419, Fall 2011
20. Print out your Fit Malus graph and attach it at the end of your lab. Only print one per group.
Note: The following experiment will use all of the setup from
Activity 2-1. Leave everything in place.
Activity 2-2: Three Polarizer Experiment
1. Using the setup from Activity 2-1, set the two existing polarizers so that they are crossed (e.g. the polarizer at 90º and
the analyzer at 0º).
Make sure to leave the heat absorbing filter in place between
the lamp and the polarizers!
Prediction 2-1: What is the orientation of the electric field after it passes through the first polarizer? What will happen to this light
when it reaches the second polarizer?
Prediction 2-2: With this arrangement, what output do you expect from the light sensor?
Prediction 2-3: A third polarizer will be added in between the other two. What effect will this have, if any, on the output of the
light sensor?
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University of Virginia Physics Department PHYS 2419, Fall 2011
Prediction 2-4: What orientation of the third polarizer (in between the first two) do you expect would produce maximum
voltage? Give the angle with respect to the first polarizer.
2. Place the third polarizer in between the other two.
3. Look through the analyzer at the lamp. Play around with the relative orientation of the middle polarizer. Record your
observations.
4. Open the experiment file named L11A2-2 Three Polarizer. There should be two digit displays; one for voltage output and
one for intensity. Press Start to activate the displays.
5. Adjust only the middle polarizer and find the orientation for
which the output shown on the computer is a maximum. Record the angle at which the maximum occurs.
Angle
6. Click on the red square to stop the data collection.
Question 2-5: Explain your findings in terms of the orientation of the electric field after the light travels through each polarizer. Why
would the angle found in step 5 produce the maximum intensity?
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University of Virginia Physics Department PHYS 2419, Fall 2011
INVESTIGATION 3: BREWSTER’S LAW
An alternative way to produce linearly polarized light is based on Brewster’s law. A wave falling on the interface between two
transparent media is, in general, partly transmitted and partly reflected. However, there is a special case in which the directions
of the refracted and reflected waves are perpendicular to each other, as shown in Figure 9.
α α
n1
Incident
Ray Reflected
Ray
n2
β Refracted
Ray
Figure 9
The component of the wave whose electric field vector E is in the
plane of the page, called the p wave, is not reflected at all but completely transmitted when the incident angle is α (called
Brewster’s angle) from the normal. The electric field of the p wave is represented by the short lines () in the figure.
Meanwhile, the reflected light contains the remainder of the wave, the component whose electric vector oscillates perpendicular to
the plane of the page. Therefore, the light that is reflected is totally polarized. This second wave is usually called the s wave.
The electric field of the s wave is represented by the dots () in the figure.
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University of Virginia Physics Department PHYS 2419, Fall 2011
The angle of incidence satisfying the condition of Brewster’s law, called Brewster’s angle, is easily obtained from Figure 9. Noting
that
2
πα β+ = (11)
and using Snell’s law ( 1 2sin sin ,n nα β= where n1 is the index of
refraction of the medium containing the incident ray, and n2 is the index of refraction of the medium containing the refracted ray), we
can show:
2
1
sin sin sintan
sin cossin
2
n
n-
α α αα
πβ αα
= = = =
. (12)
In the case that you will be looking at in class, the index of refraction of the first medium, n1 is equal to the index of refraction
of air. For this workshop, this will be taken to be unity. Putting this into Equation (12), we get:
tann α= (13)
where n is the index of refraction of the glass plate.
Brewster’s law is just a special case of the Fresnel equations that
give the amplitudes of the transmitted and reflected waves for all angles for the two polarizations.
The polarization upon reflection is rarely used to produce polarized light since only a few percent of the incident light are reflected by
transparent surfaces and become polarized (metal surfaces do not polarize light on reflection). But the fact that light reflected by
glass, water, or plastic surfaces is largely polarized enables one to cut down glare with Polaroid glasses or Polaroid photographic
filters.
α
α
β
β
n
Figure 10
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University of Virginia Physics Department PHYS 2419, Fall 2011
If one shines light at the Brewster angle onto a plane parallel glass plate, as shown in Figure 10, the Brewster condition is satisfied at
both the entrance and the exit face. This means that the p wave is perfectly transmitted (without reflection) by both surfaces. Such
an arrangement is called a Brewster window. Such windows are often used in gas lasers. As a result, the light from these lasers is
strongly linearly polarized.
In this investigation, a laser will be used to test Brewster’s Law. For this investigation, you will need the following:
• Laser
• Polarizer
• Glass plate taped to mount
• Goniometer [Basically two sticks pinned together and a protractor to measure the angle between them. Today we won’t be using the sticks, just the protractor.]
• Small support stand
WARNING: LASER LIGHT CAN DAMAGE THE EYES.
NEVER LOOK DIRECTLY INTO THE BEAM OR AT
LASER LIGHT REFLECTED FROM METAL, GLASS OR
POLISHED SURFACES.
Activity 3-1: Determination of Brewster’s Angle
Prediction 3-1: You will be using crown glass as your Brewster
window in the following experiment. What angle do you expect to find, knowing the index of refraction of crown glass (see
Appendix A)?
1. Place the goniometer on the table next to the middle of the optical bench. Place the mounted glass plate on the goniometer
hinge. This will serve as your Brewster window. See Figure 11.
Figure 11
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University of Virginia Physics Department PHYS 2419, Fall 2011
2. Adjust the laser so that the beam produced is horizontal and is incident upon the glass plate. The distance between the laser and
glass plate should be about 50 cm, but this does not require a measurement.
3. Place a polarizer and holder on the table in between the laser and the glass plate, so the light travels through the polarizer.
Recall that only the s wave (electric field vector E parallel to the glass plate) is reflected at Brewster's angle (see Figure 9). If the
s wave is not present in the incident light, then the Brewster’s angle can be found quite easily; it will be the point at which no
light is reflected. We want to use the polarizer to only allow the p wave (electric field vector E in horizontal plane) to be incident
upon the glass plate.
Question 3-1: At what angle should we set the polarizer to
transmit only the p wave?
Polarizer angle for only p wave transmission:
Explain how you decided upon this angle:
Note: Make sure that the polarizer does not completely block the laser light. To check this, look at the glass plate to ensure that there is light incident upon it. Also try to find the refracted beam. No
matter what the angle of the glass plate is with respect to the beam, there will always be a refracted (transmitted) beam – the p wave is
always refracted.
4. Set the polarizer at the angle you just determined in
Question 3-1.
5. You may find that your laser is polarized (some of our lasers
are, some aren’t). If so, you may have to rotate the laser about its axis so that enough of the beam passes through the polarizer
to clearly visible.
6. Let the reflected light fall upon a notebook size piece of white
paper as a screen to show the reflected ray, rotate the glass plate until you find the position at which the intensity of the reflected
light becomes a minimum. You may find that by alternately
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University of Virginia Physics Department PHYS 2419, Fall 2011
“tweaking” the polarizer angle and the plate angle, you can make the reflected ray completely vanish.
Note: Be careful to position the laser beam so that it does not miss the glass plate as you rotate the spectrometer stand. Do not let the
laser beam shine into the computer screen or the light sensor.
7. Read the angle on the goniometer for the position for which the light is a minimum.
θ
8. Rotate the glass plate until the beam is reflected back into the
laser. Read the angle on the goniometer again and consider this your zero angle.
0θ
9. Find the value for αθθ =− 0 . This is your Brewster’s Angle.
α
Question 3-2: Does your experimental value of the Brewster’s
angle agree with your Prediction 3-1? If not, explain.
INVESTIGATION 4: OTHER METHODS FOR POLARIZING AND
DEPOLARIZING
DEPOLARIZATION
To change polarized light into unpolarized light one must
introduce random phase differences between the two components of the electric vector. This can be accomplished by interposing a
material that is both inhomogeneous and anisotropic across the wave front.
BIREFRINGENCE
Most of the transparent materials that one encounters daily, such as
glass, plastics, and even crystalline materials such as table salt, are optically isotropic, i.e. their index of refraction is the same in all
directions.
Some materials, however, have an optically favored direction. In
these materials the index of refraction depends on the relative
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University of Virginia Physics Department PHYS 2419, Fall 2011
orientation of the plane of polarization to that preferred direction. Such materials are called birefringent or doubly refracting.
The best known example of a birefringent material is calcite (CaCO3). Normally optically isotropic materials, such as glass,
can be given a preferred direction (and thus made to be birefringent) by stressing or bending them.
1
2
o-ray
e-ray
Figure 12
Consider a light wave traversing a birefringent crystal, as shown in
Figure 12, where the direction of propagation of the wave is
entering the crystal perpendicularly. An initially unpolarized light
beam will split into two separate linearly polarized beams. One of
these is called the ordinary ray or o-ray and the other the
extraordinary ray or e-ray. The behavior of the o-ray is
essentially that of a ray in an isotropic medium: it is refracted in
accordance with Snell’s law, and its refractive index no is
independent of the direction of travel.
The e-ray, on the other hand, behaves in a most peculiar way. Its
index of refraction ne depends on the orientation of the crystal.
Moreover, its direction of travel, after entering the crystal is not
consistent with Snell’s law. As Figure 12 shows, it will be
refracted even if its angle of incidence is 90°. On leaving the
crystal it becomes again parallel to the direction of incidence but
displaced with respect to the incident beam. Since the two
emerging rays are linearly polarized along mutually perpendicular
directions, doubly refracting crystals make very effective
polarizers: If one cuts a birefringent crystal so that the e-ray, but
not the o-ray, is totally reflected at the exit face one can produce
light that is 99.999% linearly polarized.
Another application of birefringence is the quarter-wave plate, a
device that can be used to convert linearly into circularly polarized
light and vice versa. Consider again Figure 12: Not only do the e-
and the o-rays have different speeds (due to the different indices of
refraction), but they also travel different distances in the crystal.
As a result they will be out of phase with respect to one another.
Through a suitable choice of the thickness of the crystal one can
arrange it that the phase difference between the two rays becomes
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University of Virginia Physics Department PHYS 2419, Fall 2011
a quarter of a wavelength, in which case a linearly polarized
incident light beam will be circularly polarized on leaving the
crystal.
The wavelength dependence of the index of refraction, although
small, lends itself to some pretty demonstration experiments. If
one places two Polaroid filters in front of a light source so that
their directions of polarization are perpendicular to each other, they
will appear dark. If one then places an object made of a
birefringent material between the crossed Polaroids, a multicolored
image of the object will become visible in the previously dark
field. The o-ray and the e-ray have traveled different optical path
lengths and their phases, upon leaving the object, will differ, the
difference being a function of the wavelength of the light. Since
the two rays are polarized in different directions they cannot
interfere with each other. The second Polaroid (the analyzer)
passes that component of each ray whose plane of polarization is
parallel to the direction of polarization of the filter. These
components have the same plane of polarization and can interfere.
Whether their interference is constructive or destructive will
depend on their phase difference and hence on their color.
In this investigation, you will use different objects and materials to
both polarize and depolarize light. You will need the following
materials:
• Polarized light demonstrator kit
• Optical bench with polarizers
• Small support stand with tripod base and lens holder
• Desk lamp
Activity 4-1: Depolarization
As you will recall from the readings above, random phase
differences may be introduced between the two components of the
electric field vector to depolarize the light.
Incident
light
P P'
wax
paper
Figure 13
Lab 11 - Polarization L11-23
University of Virginia Physics Department PHYS 2419, Fall 2011
1. Set up two polarizers with the desk lamp and heat-absorbing
filter as done in Investigation 2 (see Figure 13). Set the
polarizer P such that E is vertical.
Prediction 4-1: With this setup (without wax paper), what do you
expect to see as you vary angle of the analyzer P' with respect to
P? (Hint: we did this in Investigation 2).
Prediction 4-2: With the wax paper added between the polarizers,
what do you expect to see as you vary angle of the analyzer P' with
respect to P?
2. Hold the piece of wax paper from the polarizing kit in between
the two polarizers.
3. Rotate the polarizer through 180º and observe (by eye) the
transmitted light.
Question 4-1: Describe the transmitted light intensity as you
rotate the polarizer.
Question 4-2: What does this show you about the polarization of
the light through the wax paper?
L11-24 Lab 11 - Polarization
University of Virginia Physics Department PHYS 2419, Fall 2011
Activity 4-2: Birefringence by the Calcite Crystal
Set the calcite crystal from the polarization kit on the dot: •
1. Hold a polarizer over the calcite and look through it at the dot.
Question 4-3: What do you observe and with and without the
polarizer?
2. Slowly rotate the polarizer until only one dot is seen. Note the
orientation of the polarizer.
θ1 (choose a relative zero angle)
3. Rotate the polarizer again until the other dot is seen. Note the
orientation of the polarizer.
θ2 (use same zero angle as before)
Question 4-4: What does this tell you about the relative
polarization of the images created by the calcite crystal?
Activity 4-3: Interference Caused by Birefringence
1. Hold the mica sample between two crossed polarizers (set at 90º
and 0º, for example) and look through the setup at the lamp.
Incident
light mica plate
P P'
Figure 14
Lab 11 - Polarization L11-25
University of Virginia Physics Department PHYS 2419, Fall 2011
2. Tilt the mica sample slowly backwards as shown in Figure 14.
Question 4-5: What do you see?
Question 4-6: Do your observations depend on the angle at which
you hold the mica?
Activity 4-4: Birefringence Due to Stress
Replace the mica with the U-shaped piece of plastic between the
crossed polarizers.
1. Look through the polarizer at the plastic.
Question 4-7: What do you observe? Do you see light?
Question 4-8: Based on your previous observations, is the light
polarized by the plastic? Why or why not?
2. Lightly squeeze the two legs of the U toward each other while
looking at the plastic through the polarizer.
L11-26 Lab 11 - Polarization
University of Virginia Physics Department PHYS 2419, Fall 2011
Question 4-9: What do you observe?
Question 4-10: What has changed about the light through the
plastic?
Comment: The strain partially orients the molecules and makes
the plastic birefringent. From such patterns engineers can locate
regions of high strain in a plastic model of a structure and then
decide whether the structure must be redesigned or strengthened in
certain places.
Question 4-11: In which corner of your plastic is there the greatest
stress?
Activity 4-5: Polarization of Scattered Light
Sunlight is scattered while passing through the atmosphere. Light
with a short wavelength is scattered more than light with a long
wavelength. This is why the sky appears blue. Light scattered by
90° is strongly polarized. You can verify this on a clear day if you
look through a Polaroid filter in the appropriate direction of the
sky.
A similar observation can be made in the laboratory by passing
laser light through a tank of water that has been clouded by
suspending some scattering material in it. At the front of the room
there should be such a tank with a laser beam should already be
passing through it.
Lab 11 - Polarization L11-27
University of Virginia Physics Department PHYS 2419, Fall 2011
1. From the side of the tank, at a right angle with respect to the
direction of the light, examine the scattered light using a
polarizer.
Question 4-12: Record your observations and use them to discuss
the polarization of the scattered light.
L11-28 Lab 11 - Polarization
University of Virginia Physics Department PHYS 2419, Fall 2011
L12-1
University of Virginia Physics Department PHYS 2419, Fall 2011
Name Date Partners
Lab 12 - INTERFERENCE
OBJECTIVES
• To better understand the wave nature of light
• To study interference effects with electromagnetic waves in
microwave and visible wavelengths
OVERVIEW
Electromagnetic waves are
time varying electric and
magnetic fields that are
coupled to each other and
that travel through empty
space or through insulating
materials. The spectrum of
electromagnetic waves spans
an immense range of
frequencies, from near zero
to more than 1030
Hz. For
periodic electromagnetic
waves the frequency and the
wavelength are related
through
c fλ= (1)
where λ is the wavelength of
the wave, f is its frequency,
and c is the velocity of light.
A section of the
electromagnetic spectrum is
shown in Figure 1.
In Investigation 1, we will use waves having a frequency of
1.05 × 1010
Hz (10.5 GHz), corresponding to a wavelength of
2.85 cm. This relegates them to the so-called microwave part of the
spectrum. In Investigation 2, we will be using visible light, which
has wavelengths of 400 - 700 nm (1 nm = 10-9
m), corresponding to
frequencies on the order of 4.3 × 1014
- 7.5 × 1014
Hz
(430 - 750 THz). These wavelengths (and hence, frequencies) differ
102
104
103
10
106
108
107
105
1010
1012
1011
109
1014
1016
1015
1013
1018
1020
1019
1017
1022 1023
1021
10-2
10-4
10-3
1
10-6
10-8
10-7
10-5
10-10
10-12
10-11
10-9
10-14 10-15
10-13
10-1
102
104 103
106
107
105
1 MHz
1 kHz
VISIBLE LIGHT
Frequency, Hz Wavelength, m
Gamma rays
X rays
Ultraviolet light
Infrared light
Short radio waves
Television and FM radio
AM radio
Long radio waves
1 nm
1 mm
1 cm
1 m
1 km
1 Å
Microwaves
1 GHz
1 THz
1 µm
10
Figure 1
L12-2 Lab 12 - Interference
University of Virginia Physics Department PHYS 2419, Fall 2011
by nearly five orders of magnitude, and yet we shall find that both
waves exhibit the effects of interference.
Electromagnetic waves are transverse. In other words, the
directions of their electric and magnetic fields are perpendicular to
the direction in which the wave travels. In addition, the electric
and magnetic fields are perpendicular to each other.
Figure 2 shows a periodic electromagnetic wave traveling in the
z-direction and polarized in the x-direction. E is the vector of the
electric field and B is the vector of the magnetic field. Study this
figure carefully. We will refer to it often.
Direction of
Propagation
Figure 2
Electromagnetic waves are produced whenever electric charges are
accelerated. This makes it possible to produce electromagnetic
waves by letting an alternating current flow through a wire, an
antenna. The frequency of the waves created in this way equals the
frequency of the alternating current. The light emitted by an
incandescent light bulb is caused by thermal motion that accelerates
the electrons in the hot filament sufficiently to produce visible light.
Such thermal electromagnetic wave sources emit a continuum of
wavelengths. The sources that we will use today (a microwave
generator and a laser), however, are designed to emit a single
wavelength. Another essential characteristic of these two sources is
that they emit radiation of definite phase. That is to say, they are
coherent.
The inverse effect also happens: if an electromagnetic wave strikes a
conductor, its oscillating electric field induces an oscillating electric
current of the same frequency in the conductor. This is how the
receiving antennas of radios and television sets work. The associated
oscillating magnetic field will also induce currents, but, at the
frequencies we will be exploring, this effect is swamped by that of
the electric field and so we can safely neglect it.
Lab 12 - Interference L12-3
University of Virginia Physics Department PHYS 2419, Fall 2011
Electromagnetic waves carry energy. The energy density at any
point is proportional to the square of the net electric field. The
intensity (what we can observe) is the time average of the energy
density. Important Note: To find the intensity of the
electromagnetic waves at any point, we must first add up (as
vectors, of course), all of the electric fields to find the net electric
field. We cannot simply add intensities. It is this property of
electromagnetic waves1 that leads to interference effects.
In this workshop you will be studying how electromagnetic waves
interfere. We will, once again, be using two small regions of the
electromagnetic spectrum: microwaves and visible light. Look at
Figure 1 to understand the relative position of microwaves and
visible light. The microwaves that you will be using in this
experiment have a frequency of 1.05 × 1010
Hz, corresponding to a
wavelength of 2.85 cm. The name microwave is to be understood
historically: In the early days of radio the wavelengths in use were
of the order of hundreds, even thousands, of meters. Compared with
these waves, those in the centimeter region, which were first used in
radar equipment during World War. II, were indeed ‘micro’ waves.
You will recall that conductors cannot sustain a net electric field.
Any externally applied electric field will give rise to a force on the
free electrons that will cause them to move until they create a field
that precisely cancels the external field (thereby eliminating the
force on the electrons). If an electromagnetic wave strikes a
conductor, the component of its oscillating electric field that is
parallel to the wire will induce an oscillating electric current of the
same frequency in the conductor. This oscillating current is simply
the free electrons in the wire moving in response to the oscillating
external electric field.
Now you will also recall that an oscillating electric current will
produce electromagnetic waves. An important thing to note about
these induced waves is that their electric fields will be equal in
magnitude and opposite in direction to the incident wave at the
surface (and inside) of the conductor.
1 This is a general property of waves, not just for electromagnetic waves.
L12-4 Lab 12 - Interference
University of Virginia Physics Department PHYS 2419, Fall 2011
Figure 3
Consider now a plane electromagnetic wave incident upon a wire.
Figure 3 schematically shows a top view of such a case. Only one
incident wave front and the resultant induced wave front are shown.
The incident wave front is shown as having passed the wire and is
traveling from the top of the figure to the bottom. The induced wave
will be of the form of an expanding cylinder centered on the wire
and, since the induced wave travels at the same speed as does the
incident wave, the cylinder’s radius is equal to the distance that the
incident wave front has traveled since it struck the wire. At the point
where the induced wave and the incident wave touch, they add
destructively as the induced wave is 180° out of phase with respect to
the incident wave.
Figure 4
Figure 4 shows the same situation, but with a number of wires all
oriented in the same direction. We can see that the induced wave
fronts all line up in phase at the incident wave front. However, since
the induced waves are 180° out of phase with the incident wave, the
resulting wave front is reduced in amplitude. We indicate this by
Direction of Propagation
of Incident Wave Front
Incident Wave
Front
Induced
Wave
Front Wire
Direction of
Propagation of Incident
Wave Front
Lab 12 - Interference L12-5
University of Virginia Physics Department PHYS 2419, Fall 2011
showing the incident wave front as a dashed line. With enough such
wires, the amplitude for the forward direction can be reduced to a
negligible level. Note that the energy of the incident wave is not lost;
it is simply re-radiated in other directions.
Figure 5
Figure 5 is similar to Figure 4 except that the wires are now arranged
in a linear array. We recognize this arrangement as the “wire grid
polarizer” from an earlier lab. In that earlier lab, we investigated the
polarization properties of the transmitted electromagnetic waves.
We now consider the properties of the scattered or reflected waves.
In Figure 5, we see that not only do the induced waves line up with
each other in the plane of the incident wave, now they also line up
with each other in another plane. This alignment of wave fronts
gives rise to constructive interference, meaning that the resulting
wave front’s amplitude is enhanced in this direction. With enough
wires, essentially all of the incident wave’s energy will be radiated in
this direction.
Furthermore, we can see from Figure 5 that the angle that this
reflected wave front makes with the plane of the wires is the same as
that of the incident wave front. In other words, “angle of incidence
equals angle of reflection”. We will find that, for microwaves, the
“wire grid polarizer” makes a fine mirror (but only for waves with
their electric fields aligned with the wires!).
What about Polaroid glasses and filters? Why do they not act like
mirrors? The answer is that Polaroid filters and glasses are thick
relative to the wavelength of visible light. The conductive molecules
are randomly distributed throughout the filter and, hence, are
arranged more like the wires shown in Figure 4 than in Figure 5.
Reflected
Wave
Front
Incident
Wave
Front
Direction of Propagation
of Reflected Wave Front
Direction of Propagation
of Incident Wave Front
L12-6 Lab 12 - Interference
University of Virginia Physics Department PHYS 2419, Fall 2011
INVESTIGATION 1: INTERFERENCE EFFECTS WITH MICROWAVES
Activity 1-1: Polarization Of Microwaves Reflected From A
Wire Grid
For this activity, you will need the following:
• Gunn diode microwave transmitter
• Microwave receiver
• Goniometer
• Wire Grid
IMPORTANT: It is imperative that you NOT peg the meter as
doing so can damage it! If you find the meter pegged,
immediately turn down the sensitivity and/or move the receiver
away from the microwave generator!
Receiver
Generator
Figure 6
1. Refer to Figure 6. Place the generator on the main arm of the
goniometer such that entrance to the “horn” is between
20-30 cm from the goniometer’s hinge (at the center of the circle).
Place the receiver on the goniometer’s shorter arm so that its horn
is about the same distance from the hinge.
2. Set the angle between the two arms the goniometer at 180° (so
that the receiver is "looking" directly at the transmitter). Set both
the receiver and the transmitter at 0° (so that the electric field of
the emitted and detected waves is oriented vertically). Turn on
the generator by plugging the AC adapter wire into the generator
and then plug the adapter into an AC outlet. Adjust the
sensitivity on the receiver to obtain a signal that is 75% of full
scale on the meter. Again, do not let the meter peg! Leave
the sensitivity at this setting for the remainder of this
Investigation.
Lab 12 - Interference L12-7
University of Virginia Physics Department PHYS 2419, Fall 2011
3. Place the wire grid so that the center pin of the goniometer
hinge fits into the recess in the bottom of the grid frame (see
Figure 7). Align the grid so that it is at 40° on the goniometer
scale.
Question 1-1: What is the angle (labeled “Angle of Incidence” in
Figure 7) between the incident microwave beam and the normal to
the plane formed by the wire grid array?
Angle of Incidence
Figure 7
Prediction 1-1: At what angle between the goniometer arms do we
expect to find a maximum detected signal? Explain.
Goniometer Angle
Note: You may find it easier to slide the receiver if you put a bit of
paper under the receiver to reduce sliding friction.
4. Slide the receiver (still attached to the short goniometer arm) until
you find the angle where the detected signal is at a maximum.
[Note: You may find it easier to slide the receiver if you put a bit
of paper under the receiver to reduce sliding friction.] Record the
angle between the two goniometer arms and the detected signal
strength.
Goniometer Angle
Angle of
Incidence Angle of
Reflectance
Wire Grid
Transmitter
Receiver
40°
L12-8 Lab 12 - Interference
University of Virginia Physics Department PHYS 2419, Fall 2011
Question 1-2: Based on your observations, does the wire grid array
behave like a mirror? Explain.
Activity 1-2: Two Slit Interference Pattern
To further observe interference with microwaves, you will need the
following:
• Gunn diode microwave transmitter
• Microwave receiver
• Goniometer
• Double slit hood
• Meter stick or tape measure, plastic ruler
CAUTION: DO NOT ALLOW THE RECEIVER’S METER
TO PEG AT ANY TIME!
Receiver
Generator
Figure 8
1. Slide the double slit hood over the generator’s horn, creating two
coherent microwave sources as shown schematically in Figure 9.
Lab 12 - Interference L12-9
University of Virginia Physics Department PHYS 2419, Fall 2011
x ≈ d sin θ
Microwave
generator
θ
θ
Horn
Double slit hood Receiver
d
Figure 9
2. Place the generator on the main arm of the goniometer such that
the hood lies directly over the goniometer’s hinge. Place the
receiver on the goniometer’s shorter arm so that the horns are
about 25 cm apart.
The signal amplitude that the receiver will detect depends on the
phase of the microwaves when they reach the receiver probe. To a
good approximation, if x ≈ d sin θ is equal to an integral number of
wavelengths nλ, then the microwaves from the two slits will
interfere constructively and you will see a maximum register on
the meter. Likewise, if d sin θ is equal to a half-integral number of
wavelengths (n - ½) λ, the meter will register a minimum.
constructive interference: sinn dλ θ=
destructive interference: ( )1 2 sinn dλ θ− =
3. Record the distance between centers of slits (double slit hood)
d: __________
4. Adjust the horn around θ = 0º to obtain a maxima signal. Then move the horn receiver to greater angles and note further
maxima and minima. You will need to increase the sensitivity to find the minima accurately. However, reduce the sensitivity as
you move away from the minima so that you do not peg the meter! You should be able to locate a minimum and a maximum
on either side of the central maximum (0º). [Remember, it is easier to slide the receiver if you place a sheet of paper under its
feet.]
Angles of minima: _____________ ______________
Angles of maxima: _____________ ______________
L12-10 Lab 12 - Interference
University of Virginia Physics Department PHYS 2419, Fall 2011
5. Use your data for the minima to find the wavelength. Show your calculations below.
λ: __________________
6. Use your data for the first non-central maxima to find the
wavelength. Show your calculations below.
λ: _________________
Question 1-3: How do your values compare with the given microwave wavelength (28.5 mm)? Discuss any uncertainties.
7. Turn off the receiver and set it aside.
Activity 1-3: Standing Waves
When waves moving in a given medium have the same frequency, it is possible for the waves to interfere and form a stationary
pattern called a standing wave. Standing waves, though they are not found in all waves, do occur in a variety of situations, most
familiarly perhaps in waves on a string, like in a guitar or violin. The incident and reflected waves combine according to the
superposition principle and can produce a standing wave.
We have seen how a grid of wires acts like a mirror for
microwaves. A metal plate can be thought of as the limit as the spacing between the wires vanishes. Microwaves reflected from a
metal plate have the same frequency and wavelength as the incident microwaves, but they travel away from the plate and their
phase is such that they add with the incident wave so as to cancel at the plate. At certain distances away from the plate (even
number of quarter-wavelengths, such as 2λ/4, 4λ/4, 6λ/4, …), the electric fields of the two waves will again destructively interfere
and produce a minimum signal in the detector probe, while at other
Lab 12 - Interference L12-11
University of Virginia Physics Department PHYS 2419, Fall 2011
locations (odd number of quarter-wavelengths, such as λ/4, 3λ/4, 5λ/4, …) they will constructively interfere and produce a
maximum signal.
Consider the configuration shown in Figure 10 (below). The
incoming field from the generator will be reflected from the metal plate and subsequently interfere with the incident wave. We can
use the detector to find the positions of the maxima and minima and determine the wavelength of the electromagnetic field.
Figure 10
For this activity, you will need the following materials:
• Gunn diode microwave transmitter
• Microwave probe
• Microwave receiver
• Metal reflector plate
• Goniometer
• Component holders
• Meter stick or tape measure, plastic ruler
1. Locate the microwave detector probe, a rectangular piece of
circuit board (and attached cord) to which a detector diode (an electrical device that conducts current in only one direction) is
soldered. Notice the solder line that extends beyond the ends of the diode and acts as an antenna.
The antenna is designed to be equal to the length of two wavelengths, i.e. 5.7 cm. When the electric field of the microwave strikes it, an ac
voltage at a frequency 10.5 GHz is induced across the diode.
The amplitude of the DC signal from the detector diode is generally
quite weak, so it must be amplified.
L12-12 Lab 12 - Interference
University of Virginia Physics Department PHYS 2419, Fall 2011
2. Plug the diode cable into the jack on the side of the microwave receiver. Make sure that the receiver is pointed away so that the
horn does not “see” any of the signal.
3. Position the diode at about 50 cm from the front of the
microwave generator’s horn and oriented vertically, as shown in Figure 10. Make sure the orientation of the generator is 0º.
4. Adjust the distance of the probe from the generator until the meter registers a voltage about 3/4 of full scale. Keep the probe
at least 15 cm away from the generator to keep the diode from burning out. [The stand holding the detector probe is
easier to slide if you put a piece of paper under its feet. Also, remember to keep your hand out of the way since any conductor
in the vicinity, e.g. a piece of metal, even your hand, will reflect waves and may give you spurious results.
Prediction 1-2: If we place a reflector behind the detector probe, the microwave should be reflected back towards the generator. What do
you think will happen to the original wave and the reflected wave? What are the conditions to produce a maximum constructive standing
wave? What are the conditions to produce a minimum?
5. Place a reflector (solid flat piece of metal) behind the detector probe, as shown in Figure 10. This will produce a standing wave
between the generator and the reflector.
6. Position the probe near the plate (at least 50 cm from the
generator) and slide it along the leg of the goniometer. Notice that there are positions of maxima and minima signal strength.
Slide the detector probe along the goniometer, no more than a cm or two, until you determine a maximum signal. Then slide the
reflector, again no more than a centimeter or two, until you obtain another signal strength maximum. Continue making
slight adjustments to the detector probe and reflector until the meter reading is as high as possible, but not pegging on the 10 V
scale. If this occurs, move the generator back further away.
7. Now find a node (minimum) of the standing wave pattern by
slightly moving the probe until the meter reading is a minimum. We want to determine the wavelength of the standing wave, so
only relative distances between maxima and/or minima are relevant. In this case, it is easiest to use the goniometer scale and
Lab 12 - Interference L12-13
University of Virginia Physics Department PHYS 2419, Fall 2011
measure the distance using the probe base and goniometer scale. Record the position of the probe below:
Initial probe position at minimum: ____________________
8. While watching the meter, slide the probe along the goniometer
until the probe has passed through at least ten antinodes (maxima) and returned to a node. Be sure to count the number
of antinodes that were traversed. Record the number of antinodes traversed and the new probe position.
Antinodes traversed: ______
Final probe position at minimum: ________
Question 1-4: What are the analogies with the nodes and antinodes found here and for the standing waves found from an oscillating
string fixed at both ends (guitar)? Sketch a picture.
Question 1-5: What is the distance in terms of the wavelength between adjacent antinodes (maxima)?
Question 1-6: What is the wavelength you deduce from your data?
Show your work.
λ : ________________________
L12-14 Lab 12 - Interference
University of Virginia Physics Department PHYS 2419, Fall 2011
Question 1-7: Using this experimental wavelength value, determine the frequency of the wave. Show your work and discuss the
agreement with what you expect.
f : ____________
9. Unplug the generator’s power supply from the receptacle and turn off the amplifier before proceeding.
INVESTIGATION 2: INTERFERENCE EFFECTS WITH VISIBLE LASER
LIGHT
Please read Appendix D: Lasers before you come to lab.
OVERVIEW
In an earlier experiment you studied various interference phenomena with electromagnetic waves whose wavelength λ was
approximately 3 cm (microwaves). In this experiment you will study similar phenomena with electromagnetic waves in the visible
part of the spectrum. For brevity, we will simply use the common term “light”. Light waves have a much shorter wavelength
(λ ≈ 4 - 7 × 10-5
cm) than do microwaves.
All the phenomena that you will observe can be described quite
accurately with a simple theoretical model dating back to Christian Huygens (1629 - 1695). This model applies to wave phenomena in
general and does not make any reference to the electromagnetic nature of light.
Huygens’ Principle states that every point of a wave front can be thought of as the origin of spherical waves. This seems to be
contradicted by experience: How can a laser emit a pencil beam? Would the light not spread out immediately?
Lab 12 - Interference L12-15
University of Virginia Physics Department PHYS 2419, Fall 2011
Figure 11
We investigate this question with a gedanken (thought)
experiment: Let a plane wave be incident on a screen that has a hole cut into it. Imagine that at the time t = 0 the wave front,
coming from below, has just reached the hole as shown in Figure 11a. An instant later, at t = t1, the spherical wavelets that
were, according to Huygens’ Principal, created at every point of this wave front have begun to spread, as in Figure 11b. Later yet at
t = t2, shown in Figure 11c, we find that the wavelets have spread considerably; but, in a central region as wide as the slit, they have
formed a new wave front propagating in the same direction as the old one. This reasoning can be repeated point by point in space
and time as needed.
As a result, one will see by and large what one would have
expected: a ray of light of the width of the slit, propagating in the original direction. Only on very close observation will one see that
a small amount of light has leaked around the corner to regions where according to a ray model it should not be. The larger the
hole, the smaller will be the fraction of the light that leaks around the corner, a process called diffraction. “Largeness” is a relative
term - large with respect to what? The only measure of length that is appropriate for this problem is the wavelength of the incident
light: When the size of the hole is large compared to the wavelength only a small part of all the light will find its way
around the edge, most of it will be in the central beam. Only if the size of the hole is small compared to the wavelength will one find
that the light spreads out spherically, as shown in Figure 12 (below). Clearly this is only a hand waving argument. A
L12-16 Lab 12 - Interference
University of Virginia Physics Department PHYS 2419, Fall 2011
mathematically rigorous explanation of diffraction, based on Huygens’ Principle, is credited to J. A. Fresnel, (1788-1827).
Figure 12
For this investigation, you will need the following:
• Laser
• Lab jack
• Small Support stand
• Clamp for stand
• Slide with single slits of varying widths
• Slide with multiple slits
• Diffraction grating slide
• White paper
• Board placed on table to observe patterns
• 3-m tape
THE SINGLE SLIT
Diffraction patterns can be observed with a single slit. Figure 13
(below) shows three representative wavelets emerging from a single slit. The angle has been chosen so that the path difference
between the wavelets a) and c) is equal to the wavelength λ. For a single slit such a condition will result in darkness: For each wavelet emerging from the lower half of the slit there will be one
from the upper half that is out of phase by half a wavelength that will extinguish it.
Lab 12 - Interference L12-17
University of Virginia Physics Department PHYS 2419, Fall 2011
Figure 13
In the case of the single slit the condition for darkness is thus
dark
n
n Dy
w
λ= (2)
where n is any non-zero integer (n = ±1, ±2, ±3…).
A more detailed derivation gives the intensity, I, as a function of
angle, θ:
2
0 2
sin sin
( )
sin
w
I = Iw
πθ
λθ
πθ
λ
. (3)
I(θ) is shown in Figure 14a. Dark bands appear when the intensity
drops to zero. It is easy to see that yndark
correspond to such
minima (since sin y Dθ = ). Bright bands appear at the maxima
of the intensity distribution. By inspection, we can see that the intensity goes to I0 as θ goes to zero (use the small angle
approximation: sin x x≈ for very small x ). The rest of these
maxima are best found through numerical methods.
L12-18 Lab 12 - Interference
University of Virginia Physics Department PHYS 2419, Fall 2011
-0.1 -0.05 0 0.05 0.1
Minima corresponding
to y2dark
Minima corresponding
to y1dark
Figure 14a Figure 14b
Activity 2-1: Single Slit Interference
CAUTION: THE LASER BEAM IS VERY INTENSE. DO
NOT LOOK DIRECTLY INTO IT! DO NOT LOOK AT A
REFLECTION OF THE BEAM FROM A METALLIC
SURFACE.
1. Aim the laser at white paper placed on the board at least 1 m away, and turn it on. Note the brightness of the spot on the
screen. The parallel beam of light is only a few millimeters in diameter.
2. Clamp the slide holder containing the single slit slide in the support stand so that the slits are horizontal.
3. Use the lab jack to adjust the height of the laser so that the beam passes through the second to narrowest single slit
(w = 0.04 mm).
4. Observe the diffraction pattern on the screen. You should see
something like shown in Figure 14b. [The black “bars” in the figure represent the bright bands while the gaps represent the
dark bands.]
5. Record your observations. Approximately how wide are the
bright bands?
2y1dark 2y2
dark
Lab 12 - Interference L12-19
University of Virginia Physics Department PHYS 2419, Fall 2011
Question 2-1: Discuss your observations in terms of Figure 14.
6. Measure the distance from the slit slide to the screen.
D:
7. Measure the diffraction pattern for this slit. To do this, measure from the dark band on one side of the center to the dark band on
the other side of the center light band (see Figure 14). This will give you 2yn
dark. Measure for three different bands and record in
Table 2-1.
Table 2-1
8. Use the values found for 2yndark to determine yn
dark and record
them in Table 2-1.
Question 2-2: Use the bands and the measurements you found to
find the wavelength of the laser using Equation (2). Record the values in Table 2-1. Find the average and record below.
λ
9. Move the laser so that it is incident on the wider slits of 0.08 mm, and then on 0.16 mm. Look at the intensity pattern for
each.
w n 2yndark
yndark
λ
0.04 mm
1
2
3
L12-20 Lab 12 - Interference
University of Virginia Physics Department PHYS 2419, Fall 2011
Question 2-3: What can you qualitatively say about how the diffraction pattern changes as the slit width goes from smaller to
larger?
Question 2-4: Why do you think that the bands would get larger or smaller (depending on your answer to Question 2-3)? Why
would the bands get closer together or further apart (also depending on your previous answer)?
THE DOUBLE SLIT
We consider now the case of two very narrow parallel slits making the following assumptions:
1. The distance d between the slits is large compared to their widths.
2. Both slits are illuminated by a plane wave e.g. light from a distant source or from a laser.
3. The incident light has a well-defined wavelength. Such light is called monochromatic, i.e. light of one color.
4. The individual slits are narrow, no more than a few hundred wavelengths wide.
Each of the two slits is the source of wavelets. Since the slits are very narrow, each is the source of just one series of concentric
cylindrical wavelets, as shown in Figure 15. Both slits are illuminated by the same plane wave, the wavelets from one slit
must, therefore, be in phase with those from the other.
Let the black rings in Figure 15 indicate the positions of positive ½
waves (maxima) at a certain moment in time. In certain directions one sees black “rays” emanating from a point half way between the
slits. In these directions the waves from the two slits will overlap and add (constructive interference). In the directions in between
Lab 12 - Interference L12-21
University of Virginia Physics Department PHYS 2419, Fall 2011
the black “rays” positive half waves (black) will coincide with negative half waves (white) and the waves from the two slits will
extinguish each other (destructive interference).
Figure 15
Figure 16 gives an example of destructive interference: At the
angle θ shown in the figure, the waves from the two slits are out of step by a half wavelength. Clearly, destructive interference will
also result in all those directions for which the waves from the two slits are out of step by an odd number of half wavelengths. In
those directions in which the distances traveled by the two waves differs by an even number of half wavelengths the interference will
be constructive.
L12-22 Lab 12 - Interference
University of Virginia Physics Department PHYS 2419, Fall 2011
Figure 16
On a screen intercepting the light, one will therefore see alternating light and dark bands. From the positions of these bands it is easy
to determine the wavelength of the light. From Figure 16 we find
sin = 2d
λθ (4)
where d is the distance between the slits and λ is the wavelength.
We can also note that
1tandark
y =
Dθ (5)
where y1dark
is the distance of the first dark band from the center
line and D is the distance from the slit to the screen. From this we
can obtain (for small angles, tan sinθ θ≈ ) the wavelength:
1
dark2y d
Dλ ≈ (6)
Convince yourself that this can be generalized to
, 1,2,3,...dark
n2y d
n(2n - 1)D
λ ≈ = (7)
where y1dark
, y2dark
, y3dark
, etc. are the distances to the first, second,
third, etc. dark band from the centerline.
Lab 12 - Interference L12-23
University of Virginia Physics Department PHYS 2419, Fall 2011
Activity 2-2: Multiple Slit Interference
1. Replace the single slit slide with the multiple slit slide.
2. Make the laser light incident upon the double slit with the slide.
Question 2-5: Discus how this pattern is different from the single
slit pattern.
3. With a pencil or pen, mark a line slightly off to the side the
position of the central ray. Then mark each of the first four dark
spots above and below the central ray position. It may be
easiest to take the white paper off the board to measure the
positions between the dark spots above and below the critical
rays. Mark on your diagram where the distances 2y1dark
, 2y2dark
,
and 2y3dark
actually are. Measure them and write them down
below.
2y1dark
___________________________
2y2dark
___________________________
2y3dark
___________________________
4. Use these measurements calculate the wavelength. Show your
work.
λ
L12-24 Lab 12 - Interference
University of Virginia Physics Department PHYS 2419, Fall 2011
Question 2-7: How well does this value agree with the single slit
determination?
THE DIFFRACTION GRATING
Adding more slits of the same widths and with the same slit to slit
distances will not change the positions of the light and dark bands
but will make the light bands narrower and brighter. In diffraction
gratings this is carried to extremes: many thousands of lines are
scratched into a piece of glass or a mirror surface, giving the same
effect as many thousands of slits. The “slit-to-slit” distance d is
usually made very small so that for a given wavelength λ the
distances ynbright
from the bright bands to the center become very
large.
At the same time, the thickness of the bands of monochromatic
light become narrow lines. This enables one to measure the
ynbright
’s (and hence the wavelengths) very accurately with such
gratings. The order m specifies the order of the principal maxima,
and m = 0 for the central beam at a scattering angle θ = 0. The first
bright spot on either side of the central maxima would be m = 1
scattered at angle θ1; the next spot would be m = 2 scattered at
angle θ2 and so forth.
It is therefore quite easy to calculate the wavelength of light using
a diffraction grating. The wavelength is given by the equation
sin mm dλ θ= (8)
D is the distance from the diffraction grating to the screen, and the
first maximum (m = 1) is observed at an angle θ1 from the central
ray. Then ymbright
is the distance on the screen from the central ray
to the maxima corresponding to order m. Then
tan bright
mm
y
Dθ = (9)
By using the two previous equations we can calculate the
wavelength. Equation (9) can be used to find θm, and then that
value plugged into Equation (8).
Lab 12 - Interference L12-25
University of Virginia Physics Department PHYS 2419, Fall 2011
Activity 2-3: The Diffraction Grating
In this activity, you will use the diffraction grating to calculate the
wavelength of the light.
1. Find the slide that contains the diffraction grating. Printed on
the slide should be the number of lines per millimeter (N).
Record this value and then determine the distance d between
adjacent slits 1/N.
N: d:
NOTE: The grating surface is about six millimeters in from the
face of the housing.
2. Measure the distance D between the diffraction grating and the
screen very precisely. It should be about 15-20 cm, to allow for
the band separations to be as large as possible while still being
on the screen.
D:
3. Shine the light from the laser through the diffraction grating and
ensure that you can measure the maxima. You probably will
have to look at the first order, because the others will be off the
screen. It may help to turn the grating and screen horizontal to
make these measurements.
4. Measure and record the distance from the center of one
maximum to the maxima on the other side of the central ray
corresponding to the same order:
2ymbright
:
5. Record the order your used:
m:
6. Use your data to calculate the wavelength of the laser light.
λ:
L12-26 Lab 12 - Interference
University of Virginia Physics Department PHYS 2419, Fall 2011
Question 2-8: Discuss how well this value for the laser
wavelength agrees with your previous values. Which method
(single slit, double slit, or grating) appears to be the best? Explain
and discuss possible sources of uncertainties for each method.
A-1
APPENDIX A
SELECTED CONSTANTS
I. Fundamental Constants Cgs Mks
Speed of light ( c ) 2.99792458* × 10
10 cm • s-1
108
m • s-1
Gravitational constant (G ) 6.67 × 10-8
dyn • cm2 • g
-2 10
-11 N • m2 • kg
-2
Permeability constant ( 0µ ) 1.26 × –– 10-6
Henry • m-1
Permittivity constant ( 0ε ) 8.85 × –– 10-12
Farad • m-1
Electron charge ( e ) 1.60219 × –– 10-19
Coulomb
4.80325 × 10-10
esu
Planck’s constant ( h ) 6.6262 × 10-27
erg • s 10-34
J • s
4.1357 × 10-15
eV • s
Planck’s constant ( π= 2hℏ ) 1.05459 × 10-27 erg • s 10-34 J • s 6.5822 × 10-16 eV • s
Avogadro’s number ( AN ) 6.022 × 1023 mol-1 1023 mol-1
Gas constant ( R ) 8.314 × 107 erg • K-1 mol-1 1 J • K-1 • mol-1
Bohr radius ( 0a ) 0.529177 × 10-8 cm 10-10 m
Rydberg ( R ) 1.09737 × 105 cm-1 107 m-1
13.6058 × eV/ hc
II. Other Physical Constants Cgs Mks
Acceleration of gravity ( g ) 9.80665 × 102 cm • s-2 m • s-2
local 9.809 × 102 cm • s-2 m • s-2
at equatorial sea level 9.78 × 102 cm • s-2 m • s-2
at polar sea level 9.83 × 102 cm • s-2 m • s-2
Earth’s radius (earth
r ) 6.38 × 108 cm 106 m
Earth’s mass (earth
m ) 5.98 × 1027 g 1024 kg
Electron rest mass (e
m ) 9.1095 × 10-28 g 10-31 kg
Proton rest mass (p
m ) 1.6726 × 10-24 g 10-27 kg
Neutron rest mass (n
m ) 1.6748 × 10-24 g 10-27 kg
Speed of sound in dry air at STP 3.31 × 104 cm • s-1 102 m • s-1
* This number is exact as the meter is now defined in terms of c and the second.
A-2
II. Other Physical Constants Cgs Mks (continued)
Heat capacity of water 4.19 × 107 erg • g-1 • K-1 103 J • kg-1 • K-1
Heat of fusion of water (at 100°C) 3.34 × 109 erg • g-1 105 J • kg-1
Heat of vaporization of water (at 0°C) 2.27 × 1010 erg • g-1 106 J • kg-1
Index of refraction ( n ) of
water (589.2 nm) 1.33
crown glass (589.2 nm) 1.52
air (590.0 nm) 1.0002765
III. Conversion Factors
1 Electron volt (eV) 1.60219 × 10-19 J
1 angstrom (Å) 0.1 nm
1 Pascal (Pa) 1 N • m-2
1 Torr 1 mm Hg = 133 Pa
1 atmosphere (atm) 760 Torr = 101.3 kPa
1 erg 10-7 J
1 calorie 4.18 J
1 Tesla 1 Weber • m-2 = 104 Gauss
IV. Standard Resistor Color Code
first three bands fourth band tolerance black 0 green 5 unmarked 20%
brown 1 blue 6 silver 10% red 2 violet 7 gold 5%
orange 3 gray 8 yellow 4 white 9
V. Temperatures of Substances (at a pressure of one atmosphere)
Boiling point of water 100°C = 373.15 K
Melting point of ice 0°C ≡ 273.15 K
Dry ice + methanol -78.5°C = 194.7 K
Boiling point liquid nitrogen -195.8°C = 77.4 K
Boiling point liquid helium -269.0°C = 4.2 K
VI. Color as Function Of Wavelength
400 - 450 nm Violet
450 - 500 nm Blue 500 - 575 nm Green
575 - 595 nm Yellow 595 - 620 nm Orange
620 - 700 nm Red
B-1
APPENDIX B
GRAPHICAL ANALYSIS
Introduction
There are two basic ways to present data: the data table and the graph. In this appendix,
we will try to acquaint you with some of the points to be considered in preparing a proper
graph. The discussion assumes hand drawn paper graphs, but the ideas are, of course,
applicable when using graphing software.
Selecting the Graph Paper
For a first graph, made while the data are being taken, you might find it convenient to just
use the square-ruled paper of your lab notebook. If you are like most people, you will be
able to divide a small length into five equal parts by eye with sufficient accuracy. A
typical notebook has squares ¼ × ¼ inch. Allowing some margins for labeling that
leaves you an area of about 30 × 40 squares. Counting on your ability to interpolate to
about 1/5 of the width of an individual square this will permit you to plot a graph
containing 150 × 200 units. Choose your scale so that the graph fills the page as much as
possible without, however, going to strange units. Thus, if you want to plot 100 seconds
along the 30 squares of the x -axis, you will be better off if you use units of 5 sec per
division instead of 3.333 sec/div, even though the latter choice would have filled the
available space. Do not use 8 div/inch because that forces you to use fractions instead of
the more convenient decimal scale.
For a formal lab report as well as for a better looking journal, you might want to use
regular graph paper ruled in either millimeters or 1/10 of an inch. Again, avoid the kind
of graph paper that is ruled in 1/2, 1/4, and 1/8 inches.
Logarithmic Paper
Sometimes it becomes necessary or desirable to plot the logarithm of a quantity instead
of the quantity itself:
Necessary: In the case that a parameter varies over several orders of magnitude, the
drastic compression of the scale by the logarithm ( log10 1= , log100 2= , log1000 3= ,
etc.) makes it possible to plot the data over a very wide range of the argument and/or the
function.
Desirable: A logarithmic plot brings out the functional relation between the two
variables plotted along the x- and y-axes. Consider, for example, the function
ay Cx= (1)
For a = 1 it is represented by a straight line, for any other value of a by some curve or
other. If we take the logarithm of both sides of this equation we get
log log logy C a x= + . (2)
This equation is represented by a straight line if we plot log x along the abscissa (the
x-axis) and log y along the ordinate (the y-axis) of linear graph paper. Doubly
B-2
logarithmic graph paper is ruled so that we do not need to take the logarithm, we simply
plot x along the (logarithmically ruled) x-axis and y along the (logarithmically ruled)
y-axis and get a straight line in the resulting log-log plot. The slope of that straight line is
given by the exponent a. Note that this is true only for a simple power law of the form
given by Equation 1. Even an additive constant will make the log-log plot non-linear.
If one wishes to plot an exponential function of the form
axy Ce= , (3)
it is expedient to use semi-log paper, i.e. graph paper on which the x-axis is ruled
linearly and the y-axis logarithmically. If one plots Equation 3 on such paper, a straight
line with the slope a will result. Again any additive term, even a simple constant will
make the resulting plot non-linear.
To find the slope from a logarithmic graph, read off two points from your straight line
and solve the relevant equation, solving for the slope. For example, working from
Equation 2,
1 1
2 2
log log log
log log log
y C a x
y C a x
= +
= +
, (4)
so
2 1 2 1log log (log log )y y a x x− = − , (5)
or
( )
( )
2 1
2 1
log
log
y ya
x x= , (6)
where x1, x2, y1, and y2 are read directly off the graph.
Just in case you do not remember what logarithms are all about, we list some useful
formulas and definitions:
ln natural log loge
x x x≡ = xxx 10loglogcommonlog =≡
( ) xe x=ln ( ) xx
=10log
ln 2.3026logx x= baab lnln)ln( +=
bab
alnlnln −=
( ) axa x lnln =
Plotting the Graph
Select scales for the x - and y -axes so that the graph fills the available area as much as is
practical. In a linear plot, you might consider suppressing zero. For example, if y varies
only from 100 to 125 you might want to start the y -axis at y = 100 and spread the
interval from 100 to 125 over the entire length of the axis. Plot each individual point first
with a pencil and, after a final check of all points, mark them permanently. A dot with a
concentric circle around it will both mark a point precisely and draw attention to it. If
you plot more than one curve on a graph, select different symbols for the data points of
B-3
each curve. Indicate the size of the probable uncertainty by error bars; if appropriate
show error bars in both the x - and y -directions.
One usually plots not only the measured data points on a graph but connects them also
with some kind of curve. This curve can either be a graphic representation of a theory
that (one hopes) describes the experiment or it can simply be a smooth line drawn by eye
with a French curve that more or less follows the data points. Whatever it is should be
clearly stated in the caption underneath the figure, e.g. “The curve shows the calculated
values according to … [the formula],” or “The curve was drawn to guide the eye.”
Often the slope of a curve, i.e. the tangent of the angle with the x -axis, conveys useful
information. This does not mean that you can learn anything by taking a protractor to
your curve, measuring its angle with the x -axis and then taking the tangent of that angle.
Let us assume that you have plotted a distance x (measured in meters) against the time t
(measured in seconds) it takes to travel that distance. In that case, the slope x t∆ ∆ of the
resulting curve will be a measure of the velocity. This slope, however, will depend on
the scales that you have used along the two axes. You will get the correct value of the
velocity in m/s only if you divide the value of x∆ (in meters) that you have read off the
y -scale by the value of t∆ (in seconds) read off the x -scale.
Labeling the Graph
Every graph should have a title that tells what is shown. You should also label both axes
and give the units used. The customary way is to give the name of the variable followed
by the dimension in square brackets, e.g.: time [sec]. A graph without proper labeling of
the axes gives no useful information.
An Example
Figures 1 and 2 (following) show an example of the same set of data plotted on a linear
and a log-log plot. The time it took a free falling steel sphere of 1 cm diameter to fall a
distance S was measured at 20 cm intervals. The experimental uncertainties were
smaller than the size of the data points. The curve represents the equation gSt /2=
for g = 9.8 m/s2.
B-4
Figure 1. Time as a function of distance in free fall.
Figure 2. Same as Figure 1, but plotted on log-log plot.
C-1
APPENDIX C
ACCURACY OF MEASUREMENTS AND
TREATMENT OF EXPERIMENTAL
UNCERTAINTY
“A measurement whose accuracy is unknown has no use whatever. It is there-
fore necessary to know how to estimate the reliability of experimental data and
how to convey this information to others.”
—E. Bright Wilson, Jr., An Introduction to Scientific Research
Our mental picture of a physical quantity is that there exists some unchanging, underlying
value. It is through measurements we try to find this value. Experience has shown that
the results of measurements deviate from these “true” values. The purpose of this Ap-
pendix is to address how to use measurements to best estimate the “true” values and how
to estimate how close the measured value is likely to be to the “true” value. Our under-
standing of experimental uncertainty (sometimes called errors) is based on the mathemat-
ical theory of probability and statistics, so the Appendix also includes some ideas from
this subject. This Appendix also discusses the notation that scientists and engineers use
to express the results of such measurements.
Accuracy and Precision
In common usage, “accuracy” and “precision” are synonyms. To scientists and engi-
neers, however, they refer to two distinct (yet closely related) concepts. When we say
that a measurement is “accurate”, we mean that it is very near to the “true” value. When
we say that a measurement is “precise”, we mean that it is very reproducible. [Of course,
we want to make accurate AND precise measurements.] Associated with each of these
concepts is a type of uncertainty.
Systematic uncertainties are due to problems with the technique or measuring instrument.
For example, as many of the rulers found in labs have worn ends, length measurements
could be wrong. One can make very precise (reproducible) measurements that are quite
inaccurate (far from the true value).
Random uncertainties are caused by fluctuations in the very quantities that we are meas-
uring. You could have a well calibrated pressure gauge, but if the pressure is fluctuating,
your reading of the gauge, while perhaps accurate, would be imprecise (not very repro-
ducible).
Through careful design and attention to detail, we can usually eliminate (or correct for)
systematic uncertainties. Using the worn ruler example above, we could either replace
the ruler or we could carefully determine the “zero offset” and simply add it to our rec-
orded measurements.
Random uncertainties, on the other hand, are less easily eliminated or corrected. We
usually have to rely upon the mathematical tools of probability and statistics to help us
determine the “true” value that we seek. Using the fluctuating gauge example above, we
C-2
could make a series of independent measurements of the pressure and take their average
as our best estimate of the true value.
Probability
Scientists base their treatment of random uncertainties on the theory of probability. We
do not have space or time for a lengthy survey of this fundamental subject, but can only
touch on some highlights. Probability concerns random events (such as the measure-
ments described above). To some events we can assign a theoretical, or a priori, proba-
bility. For instance, the probability of a “perfect” coin landing heads or tails is 12
for each
of the two possible outcomes; the a priori probability of a “perfect” die* falling with a
particular one of its six sides uppermost is 16
.
The previous examples illustrate four basic principles about probability:
1. The possible outcomes have to be mutually exclusive. If a coin lands heads, it
does not land tails, and vice versa.
2. The list of outcomes has to exhaust all possibilities. In the example of the coin
we implicitly assumed that the coin neither landed on its edge, nor could it be
evaporated by a lightning bolt while in the air, or any other improbable, but not
impossible, potential outcome. (And ditto for the die.)
3. Probabilities are always numbers between zero and one, inclusive. A probability
of one means the outcome always happens, while a probability of zero means the
outcome never happens.
4. When all possible outcomes are included, the sum of the probabilities of each ex-
clusive outcome is one. That is, the probability that something happens is one. So
if we flip a coin, the probability that it lands heads or tails is 1 12 2
1+ = . If we toss
a die, the probability that it lands with 1, 2, 3, 4, 5, or 6 spots showing is 1 1 1 1 1 16 6 6 6 6 6
1+ + + + + = .
The mapping of a probability to each possible outcome is called a probability distribu-
tion. Just as our mental picture of there being a “true” value that we can only estimate,
we also envision a “true” probability distribution that we can only estimate through ob-
servation. Using the coin flip example to illustrate, if we flip the coin four times, we
should not be too surprised to get heads only once. Our estimate of the probability distri-
bution would then be 14
for heads and 34
for tails. We do expect that our estimate would
improve as the number of flips† gets “large”. In fact, it is only in the limit of an infinite
number of flips that we can expect to approach the theoretical, “true” probability distribu-
tion.
* …one of a pair of dice.
† Each flip is, in the language of statistics, called a trial. A scientist or engineer would probably
say that it is a measurement or observation.
C-3
A defining property of a probability distribution is that its sum (integral) over a range of
possible measured values tells us the probability of a measurement yielding a value with-
in the range.
The most common probability distribution encountered in the lab is the Gaussian distri-
bution. The Gaussian distribution is also known as the normal distribution. You may
have heard it called the bell curve (it is shaped somewhat like a fancy bell) when applied
to grade distributions.
The mathematical form of the Gaussian distribution is:
2 2
2
21
2( ) d
GP d e− σ
πσ
= (1)
The Gaussian distribution is ubiquitous because it is the
end result you get if you have a number of processes,
each with their own probability distribution, that “mix
together” to yield a final result. We will come back to
probability distributions after we've discussed some sta-
tistics.
Statistics
Measurements of physical quantities are expressed in numbers. The numbers we record
are called data, and numbers we compute from our data are called statistics. A statistic is
by definition a number we can compute from a set of data.
Perhaps the single most important statistic is the mean or average. Often we will use a
“bar” over a variable (e.g., x ) or “angle brackets” (e.g., x ) to indicate that it is an aver-
age. So, if we have N measurements i
x (i.e., 1x , 2x , ..., N
x ), the average is given by:
1 2
1
1( ... ) /
N
N i
i
x x x x x N xN
=
≡ = + + + = ∑ (2)
In the lab, the average of a set of measurements is usually our best estimate of the “true”
value*:
x x≈ (3)
In general, a given measurement will differ from the “true” value by some amount. That
amount is called a deviation. Denoting a deviation by d , we then obtain:
i i i
d x x x x= − ≈ − (4)
Clearly, the average deviation ( d ) is zero (to see this, take the average of both sides). It
is not a particularly useful statistic.
A much more useful statistic is the standard deviation, defined to be the “root-mean-
square” (or RMS) deviation:
* For these discussions, we will denote the “true” value as a variable without adornment (e.g., x).
0
C-4
2 2
1
1( )
N
x i
i
d x xN
σ
=
= = −∑ (5)
The standard deviation is useful because it gives us a measure of the spread or statistical
uncertainty in the measurements.
You may have noticed a slight problem with the expression for the standard deviation:
We don't know the “true” value x , we have only an estimate, x , from our measurements.
It turns out that using x to instead of x in Equation (5) systematically underestimates the
standard deviation. It can be shown that our best estimate of the “true” standard devia-
tion is given by the sample standard deviation:
2
1
1( )
1
N
x i
i
s x xN
=
= −
− ∑ (6)
To illustrate some of these points, consider the following: Suppose we want to know the
average height and associated standard deviation of the entering class of students. We
could measure every entering student and simply calculate the average. We would then
simply calculate x and x
σ directly. Tracking down all of the entering students, however,
would be very tedious. We could, instead, measure a representative* sample and calcu-
late x and x
s as estimates of x and x
σ .
Modern spreadsheets (such as MS Excel) as well as some calculators (such as HP and
TI) also have built-in statistical functions. For example, AVERAGE (Excel) and x (cal-
culator) calculate the average of a range of cells; whereas STDEV (Excel) and xs (calcu-
lator) calculate the sample standard deviations.
Standard Error
We now return to probability distributions. Consider Equation (1), the expression for a
Gaussian distribution. You should now have some idea as to why we wrote it in terms of
d and σ. Most of the time we find that our measurements (xi) deviate from the “true” val-
ue (x) and that these deviations (di) follow a Gaussian distribution with a standard devia-
tion of σ. So, what is the significance of σ? Remember that the integral of a probability
distribution over some range gives the probability of getting a result within that range. A
straightforward calculation shows that the integral of PG [see Equation (1)] from -σ to +σ
is about 23
. This means that there is probability of 23
for any single† measurement being
within ±σ of the “true” value. It is in this sense that we introduce the concept of standard
error.
Whenever we report a result, we also want to specify a standard error in such a way as to
indicate that we think that there is roughly a 23
probability that the “true” value is within
* You have to be careful when choosing your sample. Measuring the students who have basket-
ball scholarships would clearly bias your results. In the lab we must also take pains to ensure that
our samples are unbiased. † We'll come back to the issue of the standard error in the mean.
C-5
the range of values between our result minus the standard error to our result plus the
standard error. In other words, if x is our best estimate of the “true” value x and xσ is
our best estimate of the standard error in x , then there is a 23
probability that:
x xx x xσ σ− ≤ ≤ +
When we report results, we use the following notation:
xx σ±
Thus, for example, the electron mass is given in the 2006 Particle Physics Booklet as
me = (9.1093826 ± 0.0000016) × 10-31
kg.
By this we mean that the electron mass lies between 9.1093810×10-31
kg and
9.1093842×10-31
kg, with a probability of roughly 23
.
Significant Figures
In informal usage the least significant digit implies something about the precision of the
measurement. For example, if we measure a rod to be 101.3 mm long but consider the
result accurate to only ±0.5 mm, we round off and say, “The length is 101 mm.” That is,
we believe the length lies between 100 mm and 102 mm, and is closest to 101 mm. The
implication, if no error is stated explicitly, is that the uncertainty is ½ of one digit, in the
place following the last significant digit.
Zeros to the left of the first non-zero digit do not count in the tally of significant figures.
If we say U = 0.001325 Volts, the zero to the left of the decimal point, and the two zeros
between the decimal point and the digits 1325 merely locate the decimal point; they do
not indicate precision. [The zero to the left of the decimal point is included because dec-
imal points are small and hard to see. It is just a visual clue—and it is a good idea to pro-
vide this clue when you write down numerical results in a laboratory!] The voltage U has
thus been stated to four (4), not seven (7), significant figures. When we write it this way,
we say we know its value to about ½ part in 1,000 (strictly, ½ part in 1,325 or one part in
2,650). We could bring this out more clearly by writing either U = 1.325×10-3
V, or
U = 1.325 mV.
Propagation of Errors
More often than not, we want to use our measured quantities in further calculations. The
question that then arises is: How do the errors “propagate”? In other words: What is the
standard error in a particular calculated quantity given the standard errors in the input
values?
Before we answer this question, we want to introduce a new term: The relative error of a
quantity Q is simply its standard error, Q
σ , divided by the absolute value of Q. For ex-
ample, if a length is known to 49±4 cm, we say it has a relative error of 4/49 = 0.082. It
C-6
is often useful to express such fractions in percent*. In this case we would say that we
had a relative error of 8.2%.
We will simply give the formulae for propagating errors† as the derivations are a bit be-
yond the scope of this exposition.
1. If the functional form of the derived quantity ( f ) is simply the product of a con-
stant (C ) times a quantity with known standard error ( x and x
σ ), then the stand-
ard error in the derived quantity is the product of the absolute value of the
constant and the standard error in the quantity:
( ) f xf x Cx Cσ σ= → =
2. If the functional form of the derived quantity ( f ) is simply the sum or difference
of two quantities with known standard error ( x and x
σ and y and y
σ ), then the
standard error in the derived quantity is the square root of sum of the squares of the errors:
2 2
( , ) or ( , ) f x yf x y x y f x y x y σ σ σ= + = − → = +
3. If the functional form of the derived quantity ( f ) is simply the product or ratio of
two quantities with known standard error ( x and x
σ and y and y
σ ), then the rel-
ative standard error in the derived quantity is the square root of sum of the squares of the relative errors:
2 2
( , ) or ( , ) / | | ( / ) ( / )f x yf x y x y f x y x y f x yσ σ σ= × = → = +
4. If the functional form of the derived quantity ( f ) is a quantity with known stand-
ard error ( x and x
σ ) raised to some constant power (a ), then the relative stand-
ard error in the derived quantity is the product of the absolute value of the constant and the relative standard error in the quantity:
( ) / | | / | |a
f xf x x f a xσ σ= → =
5. If the functional form of the derived quantity ( f ) is the log of a quantity with
known standard error ( x and x
σ ), then the standard error in the derived quantity
is the relative standard error in the quantity:
( ) ln( ) /f xf x x xσ σ= → =
* From per centum, Latin for “by the hundred”.
† Important Note: These expressions assume that the deviations are small enough for us to ignore
“higher order” terms and that there are no correlations between the deviations of any of the quan-
tities x , y , etc.
C-7
6. If the functional form of the derived quantity ( f ) is the exponential of a quantity
with known standard error ( x and x
σ ), then the relative standard error in the de-
rived quantity is the standard error in the quantity:
( ) /x
f xf x e fσ σ= → =
7. A commonly occurring form is one the product of a constant and two quantities
with known standard errors, each raised to some constant power. While one can successively apply the above formulae (see the example below), it is certainly easier to just use:
22
( ) /ya b x
f
baf x Cx y f
x y
σσσ
= → = +
And, finally, we give the general form (you are not expected to know or use this equa-
tion; it is only given for “completeness”):
22
2 2 2( , ,...) ...f x y
f ff x y
x yσ σ σ
∂ ∂ → = + +
∂ ∂ (7)
Standard Error in the Mean
Suppose that we make two independent measurements of some quantity: x1 and x2. Our
best estimate of x, the “true value”, is given by the mean, 11 22
( )x x x= + , and our best es-
timate of the standard error in x1 and in x2 is given by the sample standard deviation,
( ) ( ) ( )1 2
2 211 22 1x x xs x x x xσ σ
−
= = = − + −
. Note that sx is not our best estimate of xσ ,
the standard error in x . We must use the propagation of errors formulas to get xσ .
Now, x is not exactly in one of the simple forms where we have a propagation of errors
formula. However, we can see that it is of the form of a constant, ( )12
, times something
else, 1 2( )x x+ , and so:
1 2
12x x xσ σ
+=
The “something else” is a simple sum of two quantities with known standard errors (x
s )
and we do have a formula for that:
1 2 1
2 2 2 2
2 2x x x x x x xs s sσ σ σ+
= + = + =
So we get the desired result for two measurements:
1
2x xsσ =
C-8
By taking a second measurement, we have reduced our standard error by a factor of 12.
You can probably see now how you would go about showing that adding third, 3x ,
changes this factor to 13. The general result (for N measurements) for the standard
error in the mean is:
1x xN
sσ = (8)
Example
We can measure the gravitational acceleration g near the Earth’s surface by dropping a
mass in a vertical tube from which the air has been removed. Since the distance of fall
(D), time of fall (t) and g are related by D = ½ gt2, we have g = 2D/t2. So we see that we
can determine g by simply measuring the time it takes an object to fall a known distance.
We hook up some photogates* to a timer so that we measure the time from when we re-lease a ball to when it gets to the photogate. We very carefully use a ruler to set the dis-
tance (D) that the ball is to fall to 1.800 m. We estimate that we can read our ruler to within ±1 mm. We drop the ball ten times and get the following times (ti): 0.6053,
The average of these times ( )t is 0.605154 seconds. Our best estimate of g is then
2
exp 2 / 9.8305g D t= = m/s2. This is larger than the “known local” value of 9.809 m/s2
by 0.0215 m/s2 (0.2%). We do expect experimental uncertainties to cause our value to be
different, but the question is: Is our result consistent with the “known value”, within ex-perimental uncertainties? To check this we must estimate our standard error.
VERY IMPORTANT NOTE: Do NOT round off intermediate results when making calculations. Keep full “machine precision” to minimize the effects of round-off errors. Only round off final results and use your error estimate to guide you as to how many dig-
its to keep.
Our expression for g is, once again†, not precisely in one of the simple propagation of
errors forms and so we must look at it piecemeal. This time we will not work it all out
algebraically, but will instead substitute numbers as soon as we can so that we can take a look at their effects on the final standard error.
What are our experimental standard errors? We've estimated that our standard error in
the distance (D
σ ) is 1 mm (hence a relative error, D Dσ , of 0.000556 or 0.0556%).
From our time data we calculate the sample standard deviation (t
s ) to be
0.000259 seconds. Recall that this is not the standard error in the mean (our best estimate of the “true” time for the ball to fall), it is the standard error in any single one of the time
measurements (i
t ). The standard error in the mean is st divided by the square root of the
* A device with a light source and detector that changes an output when something comes be-
tween the source and detector. † Refer to the discussion of the standard error in the mean.
C-9
number of samples (10): / 10t tsσ = = 0.0000819 seconds (for a relative error, /t tσ ,
of 0.000135 or 0.0135%).
We see that the relative error in the distance measurement is quite a bit larger than the
relative error in the time measurement and so we might assume that we could ignore the
time error (essentially treating the time as a constant). However, the time enters into g
as a square and we expect that that makes a bigger contribution than otherwise. So we don’t (yet) make any such simplifying assumptions.
We see that our estimate of g (which we denote by exp
g ) is of the form of a constant (2)
times something else ( 2/D t ) and so:
2exp /
2g D tσ σ=
2/D t is of the form of a simple product of two quantities ( D and 2
t ) and so:
( ) ( )2 2
222 2
// / / /DD t t
D t D tσ σ σ= +
Now we are getting somewhere as we have D Dσ (0.000556). We need only
find 2
2/
ttσ .
2t is of the form of a quantity raised to a constant power and so:
2
2/ 2 / 0.000271
ttt tσ σ= =
Now we can see the effect of squaring t : Its contribution to the standard error is doubled.
Consider the two terms under the square root:
( ) ( )2
22 7 2 8/ 3.09 10 and / 7.33 10D t
D tσ σ− −
= × = ×
Now we can see that, even though the time enters as a square, we would have been justi-
fied in ignoring its contribution to the standard error in g. Plugging the numbers back in,
we finally get exp
0.00608g
σ = m/s2.
We see that our result is 3.5 standard deviations larger than the “known value”. While
not totally out of the question, it is still very unlikely and so we need to look for the
source of the problem. In this case we find that the ruler is one of those with a worn end.
We carefully measure the offset and find the ruler to be 5.0 mm short. Subtracting this
changes D to 1.795 m and gexp to 9.803 m/s2, well within our experimental error*.
* The implied error in our measurement of the offset (0.05 mm) is much smaller than the error in
the original D and so we can afford to ignore its contribution to the standard error in gexp.
C-10
Summary of Statistical Formulae
Sample mean: 1
1 N
i
i
x xN
=
= ∑
(best estimate of the “true” value of x , using N measurements)
Sample standard deviation: 2
1
1( )
1
N
x i
i
s x xN
=
= −
− ∑
(best estimate of error in any single measurement, i
x )
Standard error of the mean: 1
x xs
Nσ =
(best estimate of error in determining the population mean, x )
Summary of Error Propagation Formulae*
Functional form Error propagation formula
1. ( )f x Cx= ........................................................ f xCσ σ=
2. ( , )f x y x y= ± .......................................... 2 2
f x yσ σ σ= +
3. ( , ) orf x y x y x y= × ................................ 2 2
/ | | ( / ) ( / )f x yf x yσ σ σ= +
4. ( ) af x x= .................................................... / | | / | |f xf a xσ σ=
5. ( ) ln( )f x x= ................................................ /f x xσ σ=
6. ( ) xf x e= ..................................................... /f xfσ σ=
7. ( ) a bf x Cx y= ..................................................
22
/yx
f
baf
x y
σσσ
= +
and the general form:
8. ( , ,...)f x y ....................................................
22
2 2 2 ...f x y
f f
x yσ σ σ
∂ ∂ = + +
∂ ∂
* These expressions assume that the deviations are small enough for us to ignore “higher order”
terms and that there are no correlations between the deviations of any of the quantities x , y , etc.
D-1
APPENDIX D
LASERS
The wavelength λ of light is related to the frequency ν of a light wave through
ν
λc
= , (1)
where c is the velocity of light. You may recall that, according to quantum mechanics, light
consists of individual particles, photons, whose energy E is connected with the frequency ν of the
light wave through:
λ
νhc
EhE == hence, , (2)
where h = 6.63 × 10-34
[J·s] is a fundamental constant of nature, Planck’s constant.
You may also recall that atoms can exist only in states with certain, well defined, energies. If an
atom is in its state of lowest energy, the ground state, one can lift it into one of the states of higher
energy, an excited state, by bombarding it with photons whose energy νhE = is exactly equal to
the energy difference E∆ between the ground state and the excited state.
Once in an excited state, an atom will usually decay rapidly to the ground state by emitting another
photon of the same energy EE ∆= . It was Albert Einstein who realized that an atom that is already
in an excited state can be de-excited by a photon of the proper energy EE ∆= . In going back down
to the ground state it will, of course, emit another photon of that same energy.
Is it, perhaps, possible to design an amplifier for light based on this process? You see how it might
work: take a large number of atoms in an excited state and shoot in a photon of just the right energy.
This photon will de-excite one of the atoms creating another photon. Now we have two photons of
the same energy which can hit two atoms and de-excite them creating two more photons which in
turn, etc. etc. This should work all the better since the secondary photons share with the original
ones not only the energy, they also have the same phase and travel in the same direction.
Normally atoms are in their ground state while the excited states are unoccupied. To make this
scheme work one needs to create a population inversion: lots of atoms in the upper and few in the
lower state.
For decades after Einstein’s discovery physicists believed that ‘one could show’ that such an
inversion could not be accomplished. According to Einstein, those atoms that can be easily excited
will easily decay to the ground state by spontaneous emission. Those that do tend to linger in the
excited state are those that are difficult to excite in the first place.
If something is really forbidden by a law of nature it just cannot happen. However, it is often the
case that it is not really forbidden, just that people have just not been clever enough. Such is the case
here. Several ways are now known to put atoms into an excited state and keep them there until they
can be de-excited by incident photons. The good example is the well-known helium-neon gas laser
and we shall discuss it briefly.
The helium atom has an excited state that is metastable: it lives about thousand times longer than
excited atomic states normally do. According to Einstein, it should be a thousand, or so, times
harder to excite than other states, and it is harder to do so ... by photons. However, it can quite
D-2
readily be excited by bombarding the He atom with electrons, as in a gas discharge. It is thus
possible to produce large numbers of metastable He atoms in the so-called 2S1 state (see Figure 1).
By the luckiest of coincidences, the neon atom has a long-lived excited state, the 3S2 state, that has
almost exactly the same energy as the metastable He state. When an excited He atom collides with a
Ne atom in its ground state, it quite often transfers its energy to the Ne atom, returning itself to the
ground state while leaving the latter in the excited 3S2 state, as shown in Figure 1. When hit by a
photon with a wavelength of 632.8 nm the Ne atom will be de-excited to the lower, but still excited
2P4 state. It is between the 3S2 and the 2P4 states of neon that the laser action takes place.
Transfer by
collisions 2S
2P4
Excitation by
collisions with
electrons
En
erg
y
Laser light
De-excitation by
spontaneous and
induced emission
λ = 632.8 nm
Ground states
Helium Neon
3S2
Figure 1 How a HeNe laser works.
Why between these two states? There are many atoms in the Ne ground state so that it will be quite
impossible to pump enough Ne atoms into the 3S2 state to achieve a population inversion. The 2P4
state of neon, being an excited state itself, is empty.
To make a laser really work one needs a good bit of amplification, i.e. the photons must encounter a
large number of atoms in the upper state. This can be accomplished by either making the laser tube
very long or, more practically, by putting a mirror at either end, making the photons bounce back
and forth many times. This requires a great deal of precision: the secondary photons are in phase
with the primary ones, and must remain so upon reflection by the mirrors. This requires that the
latter are an integer number of wavelengths apart. In other words, a laser tube is a highly precise
interferometer.
1The labels 2S, 3S2, etc. give the experts detailed information about the quantum state.