Experiments for Wave Lab
c
,
Northeastern University
Contents
1 Mass On a Spring
1
2 Precision Period of a Pendulum
4
3 The Mechanical Oscillator
11
4 Coupled Electrical Oscillators
18
5 Pulses on a Coaxial Cable
27
6 Speed of Light
34
7 Diffraction and Interference
46
8 Polarization
52
i
Preface
This lab manual was given to me by Jackie Krim. I believe that she got it from N. Jaggi,
and he got it from Marv Gettner, and so on back to the depths of darkness within the
physics department. I redrew the diagrams and put it into LATEX form.
Most of the experiment on the speed of light is originally from the description provided
with the apparatus by PASCO, the manufacturer.
George Alverson
ii
Safety Matters
The lasers available in the lab are all rated as having a typical power output of 0.5 mW and
are classified by the United States Center for Devices and Radiological Health (CDRH)
as Class II devices. These are generally regarded as safe for normal use, but there are
certain precautions which must be taken. First, do not stare into the beam. Extended
exposure increases the chance of damage to the eye. Second, be particularly careful when
using lasers in conjunction with light-concentrating lenses.
iii
Laboratory Instrumentation
One needs to understand the idiosyncrasies of the measuring instruments in the lab to
get the best results. Many of these quirks and tics are discussed in the individual labs,
but a few tricks with oscilloscopes and lasers are covered here.
• Your major difficulty in making a measurement with an oscilloscope may be getting
it to trigger correctly, but your problems don’t end there. The accuracy of your
measurement will depend on the accuracy of the scope calibration (and we don’t
calibrate them very often). In addition, even at the best of times, there is about a
3% error in both voltage and time measurements.
• Lasers are not instant-on devices. You will see fluctuations in the power output
on the order of 20% for the first few minutes after powering the laser up. If you
make the mistake of using a non-polarized laser with a polarizing filter, 100% power
fluctuations are possible.
In general, if you have any questions about a specific piece of apparatus, try and find the
manufacturer’s manual.
iv
Experiment 1
Mass On a Spring
I. Purpose
In this experiment you will study the simplest possible oscillating system: a mass on a
spring — with the simplest of apparatus. Yet if you work carefully, your results will be
sufficiently accurate to clearly demonstrate the inaccuracy of the usual treatment in an
introductory text.
II. Measurements
You will measure the spring constant in two ways, one static and one dynamic, and
compare the results. Further, you will measure the relation between the mass and the
period of the oscillations and compare this to the theoretical relation.
III. Apparatus
You will be provided with a stand, spring, meter stick, masses and a stop watch. You
will need to supply your own graph paper and a ruler.
IV. Procedure
IV.A Static Determination of the Spring Constant.
Take the data necessary to determine the relationship between the elongation of the
spring and the force exerted on it. Make these measurements over as wide a range of
forces as possible with the apparatus you have.
Estimate the errors in your measurements and for these errors state their type, i.e.,
systematic or random. Plot your data as you take it in order to check for gross errors
and obvious evidence that the spring is departing from the expected behavior. From
this plot, determine the value of the spring constant. (Note: After you have finished the
1
2
EXPERIMENT 1. MASS ON A SPRING
experiment, you may want to make a more careful plot and find a new better value of k,
but you need to have a value while you are doing the experiment.)
IV.B Dynamic Determination of the Spring Constant.
1. For each value of the mass, measure the time elapsed for some large number of
periods using the stop watch. Read the stop watch as accurately as possible.
Estimate the measurement error of the stop watch. Repeat each measurement ten
times in order to check for the presence of random errors. Take these measurements
for about six different masses within the range of masses used in part IV.A.
2. For each value of the mass, calculate the mean value of the period, the standard
deviation of these measurements, and the error in the mean value. Also, calculate
the period you expect based on the simple theory of the mass on a spring using
your value of k from part IV.A. Do your measured and expected values agree?
3. Make a plot of (T )2 as a function of m where T is the mean period. Should this
graph be linear? Should it go through the origin? If not, why not? Determine a
value of k from this graph. How does it compare with your results from part IV.A?
V. Analysis
In your report you should include at least the following steps.
1. A plot of the data from part IV.A. Plot elongation versus force. On the graph show
the estimated errors in the elongation. Use a ruler and your judgment to draw a
“best fit” straight line through the data. From the slope of this line determine a
value of k, the spring constant.
By drawing other straight lines through the data which represent extreme fits, you
can determine extreme values of k. Convert these to the form k ± ∆k. If you wish,
you may also do this fitting numerically to obtain a true “least squares fit” to the
data. In this way you can determine best values of k and the error in k. If you do
this, be sure to compare these values to the ones you obtained from your “eyeball”
fit.
2. Determine T and σT for each mass used in the period measurement. Using these
values, compute (T )2 and its error and plot these values as a function of the mass.
Use the least squares technique to determine the parameters for the linear relationship between these two variables. Discuss the physical significance of these
parameters and compare them with the expected value. Determine the errors in
the parameters. Using the χ2 test, determine how well your data are fit by a linear
relationship.
VI. THEORY
3
3. Compare the two values of k found in this experiment. Do they agree within
their errors? Does the system behave according to the relationships given in most
elementary texts? If not, discuss possible causes of the discrepancies.
VI. Theory
Your should know or be able to find all the background information necessary to interpret the results of this experiment. Before doing the experiment, you should find the
relationships need to determine the spring constant from its elongation and from the
period of the oscillations.
VII. Hints
The most obvious problem you will encounter is that the simple theory will not give
the period that you observe. Of course, the simple theory always begins by assuming
that the spring has no mass, while your spring obviously has mass. Further, the obvious
solution to this problem is to correct the simple theory so that the mass of the spring is
taken into account. Can you do this?
Experiment 2
Precision Period of a Pendulum
I. Purpose
In this experiment you will study the simple physical pendulum. Using electronic instruments, you will be able to measure its period sufficiently accurately to verify that it is
not a constant independent of its amplitude.
II. Measurements
You will accurately measure the period of a pendulum as a function of its amplitude
and compare your results to the calculated values, both the values given by the simple
approximation and by a more complete treatment.
III. Apparatus
You will use a pendulum, an optical switch, an electronic interval timer, and various
meter sticks, micrometers, and other measuring devices. You will also need the usual
graph paper and a ruler.
III.A Pendulum
The pendulum is a simple metal bar suspended on a knife edge which provides an essentially frictionless pivot. The amplitude θ0 of the oscillations is determined by measuring
the initial displacement of the pendulum from the vertical using a meter stick. (See
Figure 2.1)
III.B Interval Timer
The interval timer contains a very accurate frequency source (a clock). When the mode
is set to pendulum, this particular timer is designed to record and display the time for
4
IV. PROCEDURE
5
b
h
a
L
Center of Mass
θ
D
Figure 2.1. Experimental Apparatus
one complete period. If the memory switch is on, additional period times are added
to the display.You should use the 0.1 ms time setting to get the best results. If recently
calibrated, the timer should be accurate to 1%.
Check the timer settings (and verify that the timer works as you expect) before taking
data. Note that with the switch settings as described above, the most significant digit
will wrap at 2.
IV. Procedure
1. Measure all the dimensions of the pendulum necessary to calculate its period. Es√
timate the errors in these measurements. The radius of gyration ( moment of
inertia/mass) for a bar is given by
k=
(a2 + b2 )/12
where a is the length of the bar and b its width. The value of g measured in the
laboratory is g = 9.80382±0.00001 m/sec2 . Before going on, calculate the expected
period of the pendulum.
2. Using the alignment jigs and levels, make sure that the pendulum and stand are
square and vertical.
3. Measure the period of the pendulum for the initial displacements of 5, 10, 15, 20,
25, and 30 cm from the vertical. Take care to avoid parallax in these measurements. Make sure that your period measurement corresponds to the first period
6
EXPERIMENT 2. PRECISION PERIOD OF A PENDULUM
after releasing the pendulum, i.e., that the timer starts counting the first time the
pendulum traverses the light beam. Remember that the pendulum crosses the light
beam twice for each complete cycle. Repeat each measurement several times to get
some idea of the reproducibility.
4. Make a crude measurement of the “damping” of the pendulum due to “air friction”.
V. Analysis
1. Compare your results with the formula
Te = T0
θ0
1
1 + sin2
4
2
where Te is experimentally measured period and θ0 is the angle shown in Figure 2.1.
To do this, plot the measured period as a function of the quantity
1
θ0
X = sin2
4
2
Since the relationship between Te and X is expected to be linear, you can easily
check this by observing if the experimental data falls on a straight line. (This is
a very general and powerful technique. Whenever possible try to plot data in a
form which will result in a linear relationship between the variables plotted. This is
useful for two reasons. First, the straight line is easily and accurately drawn with a
straight edge while other functions are not; agreement or discrepancies of the data
from the expected relationship can easily be seen without elaborate calculations.
Secondly, if the data obey a linear relationship, the two parameters of the straight
line, the slope and intercept, can easily be found from the graph.)
2. Determine T0 independently from the slope and intercept of your plot. Estimate
the error in each of these determinations. Which is more accurate? Why?
3. Calculate the uncertainty in calculated value of T0 arising from errors in g, h, and
k 2 . Derive the following expressions:
∆T0 =
g
T0 ∆g
2 g
T0 h2 − k 2
h
2 h2 + k 2
T0
k2
∆T0 2 =
k
2 h2 + k 2
1
∆k 2 2∆a
=
2
k a
a 1 + b2 /a2
∆T0 =
δh
h
∆k 2
k2
2∆a
≈
a
VI. THEORY
7
2∆b b
2b ∆b
∆k 2 1
=
≈
≈0
2
2
2
k b
a a 1 + b /a
a2
The last two expressions assume b << a and ∆b
<< 1. Evaluate the five expressions
b
for your data and compute the total error expected in the calculated value of T0 .
4. Compare the calculated value with the measured value of T0 . Comment on the
agreement or lack thereof.
5. Estimate the change in T0 due to air friction. Is this an important effect? How can
you take it into account?
VI. Theory
VI.A Results
The period of a physical pendulum, derived using the small angle approximation, is
T0 = 2π
h2 + k 2
gh
where h is the distance between the pivot and the center of mass and k is the radius of
gyration.
a2 + b2
k2 =
12
where a and b are the length and width of the pendulum. When the period is calculated
without making the small angle approximation, the period is
Te = T0
1
θ0
1 + sin2
4
2
9
θ0
+
sin4
64
2
+ ...
VI.B Derivation of Simple Results
When the pendulum shown in Figure 2.1 is at an angle θ, the restoring torque on it is
τ = hmg sin θ
Thus the differential equation for its angular motion is
hmg sin θ = −I θ̈
where I is the moment of inertia about the pivot. The small angle approximation consists
of observing that as long as θ is small, sin θ ≈ θ so that
hmg θ = −I θ̈
8
EXPERIMENT 2. PRECISION PERIOD OF A PENDULUM
This is the usual equation for simple harmonic motion, with the corresponding period,
T0 = 2π
I
.
hmg
(2.1)
Using the parallel axis theorem, I can be written as
I = m(h2 + k 2 )
where k is the radius of gyration of the bar (mk 2 is the moment of inertia of the bar
about its center of mass). Thus the period is
T0 = 2π
h2 + k 2
gh
(2.2)
VI.C Derivation of the Exact Result
We start by writing down the total energy of the pendulum shown in the figure.
1
1
E = K + U = I θ̇2 + mgy = I θ̇2 + mgh(1 − cosθ)
2
2
(2.3)
Two comments are useful at this point:
• In calculating the potential energy term, we are treating the physical pendulum
located at its center of mass, while when calculating the kinetic energy term we are
treating it as a physical pendulum. This is perfectly correct.
• The small angle approximation can be made at this point by using the approximation
θ2
1 − cos θ ≈ 1 − 1 +
−...
2
This converts Eq(3) into
1
1
E ≈ I θ̇2 + mghθ2
2
2
This is the alternate form of the differential equation for SHM, and we can write
down the period directly; it is, of course,
T0 = 2π
I
mgh
When the pendulum is at its maximum angular position, θ0 , the kinetic energy is
zero and the potential energy is
U = mgh(1 − cos θ0 )
VI. THEORY
9
Thus the complete equation for the pendulum is
1 2
I θ̇ + mgh (1 − cos θ) = mgh (1 − cos θ0 )
2
(2.4)
This is a difficult equation, but all we want to find is the period, not the exact form of
the motion. The procedure is not obvious. We rearrange this equation using Eq.(2) to
obtain
2
dθ
2mgh
(cos θ − cos θ0 )
=
dt
I
dθ √ 2π cos θ − cos θ0
= 2
dt
T0
√
dθ
2π 2
dt = √
T0
cos θ − cos θ0
This is true for all time, so we can integrate both sides of it over the same time
interval and still have an equation. In particular, we are going to integrate this over one
full period, t ranging from 0 to T :
√ dθ
2π 2 T
√
dt =
(2.5)
T0 0
period
cos θ − cos θ0
The left hand side of this integration is trivial:
√
√ T
2π 2 T
dt = 2π 2
T0 0
T0
The right hand side is more challenging. First we must remember that as t ranges from
0 to T , θ ranges from θ0 to −θ0 and back to θ0 again. Next we rewrite the right hand
side several times:
2 θ
cos θ = 1 − 2 sin
2
cos θ0 = 1 − 2 sin
2
cos θ − cos θ0 =
2 sin2 θ0 − 2 sin2 θ
2
2
θ0
2
=
√
2 sin
θ0 1 −
2
sin2
sin2
θ
2
θ0
2
We make the following change of variables:
θ0
x = sin
2
sin φ =
θ
2
sin θ20
sin
1
θ
= sin
x
2
(2.6)
10
EXPERIMENT 2. PRECISION PERIOD OF A PENDULUM
in order to obtain
√
cos θ − cos θ0 =
√
2x 1 − sin2 φ =
Next:
2x cos φ
θ
1
cos
dθ
2x
2
so
dθ =
2x cos φdφ
cos
θ
2
= cos φdφ
2x cos φdφ
2x cos φdφ
=
= 1 − x2 sin2 φ
1 − sin2 θ2
so finally
√
1
dθ
2x cos φdφ
2 dφ
√
=√
=
cos θ − cos θ0
2x cos φ 1 − x2 sin2 φ
1 − x2 sin2 φ
Now substitute this into Eq.(6):
√
√ T
2 dφ
2π 2 =
T0
period
1 − x2 sin2 φ
(2.7)
Inspecting Eq.(7), we see that when θ = θ0 , φ = π/2 and when θ has returned to
θ0 again, θ will have increased by 2π, so the limits of integration in Eq.(8) are π/2 and
5π/2. We expand the denominator using the binomial expansion, obtaining
T
x2
3
2
sin2 φ + x4 sin4 φ + . . .
2π = π dφ 1 +
T0
2
8
2
5π
Using a table of integrals, if necessary,
T
x2 π 3x4 3π
2π = (2π +
+
( ) + ...
T0
2
8 4
T = T0
T = T0
x2 9x4
+
+ ...
1+
4
64
θ0
1
1 + sin2
4
2
θ0
9
sin4
+
64
2
+ ...
Experiment 3
The Mechanical Oscillator
I. Purpose
To study the free and forced oscillations of a mass connected to a spring.
II. Theory
The mechanical oscillator in this experiment consists of a mass connected to two springs,
as shown in Figure 3.1:
O
1
0
D
m
B cos ω t
2
x
0
x
d0
Figure 3.1. Mass with Attached Springs
Point O is held stationary during the experiment, while point D is either held fixed
(for the case of free oscillations) or is moved back and forth in simple harmonic motion
(for the case of driven oscillations). The equations describing the motion of the mass
will be derived for the cases of free and driven oscillation in the following sections.
II.A Free Oscillations
The case of free oscillation is described in detail in [French]. In summary, the equation
of motion of a free oscillator which is exposed to a damping force bv may be written as:
11
12
EXPERIMENT 3. THE MECHANICAL OSCILLATOR
(see Eqs.(3-29) and (3-30))
d2 x
dx
+ ω02 x = 0
+γ
(3.1)
2
dt
dt
k
b
ω02 =
(3.2)
γ=
m
m
where x is the position of the oscillator, m its mass and k the coefficient of the restoring
force, Fr = −kx. For a weak damping force, the solution to this equation of motion is:
(see Eqs. (3-33) and (3-34)
γ
x = Ae− 2 t cos(ωt + α)
(3.3)
k
b2
γ2
=
−
(3.4)
4
m 4m2
The presence of the damping force thus slightly changes the period from its value in the
absence of any damping. The more pronounced effect is on the amplitude of oscillation
which is constantly reduced by the factor e−γt/2 . Note that maximum amplitudes occur
when cos(ωt + α) = ±1 and that successive maxima are given by:
ω 2 = ω02 −
γ
xmax = Ae− 2 nT
where T is the period of oscillation and n is an integer describing the number of periods
elapsed. The ratio of two successive maxima is given by:
γ
xmax,n
= e 2 T ≈ eπγ/ω ≈ eπγ/ω0
xmax,n+1
Thus the decrease in amplitude occurring in one period depends on the ratio ω0 /γ. Since
many other properties of the mechanical oscillator depend on this ratio, it has been given
a special name. It is called the “Q” of the system. Values of Q are typically between 5
and 50 for simple mechanical systems.
Now let’s examine the case at hand. For free oscillations, Point D will remain fixed
in place. Let us assume that the spring constants of each of the two springs are equal
(k1 = k2 = ks ). Referring to Figure 3.1:
• Let x0 be the distance from point O to the mass when it is in its equilibrium
position.
• Let x1 be the distance from point O to the mass at any instant of time.
• Let F1 = −ks (x1 − x0 ) be the force on m due to the first spring.
• Let F2 = −ks (x1 − x0 ) be the force on m due to the second spring.
The total restoring force on the mass (F1 + F2 ) is therefore −2ks (x1 − x0 ). Identifying
the restoring force with Fr = −kx assumed in Eqs.(1) and (2), we see that Eqs.(3) and
(4) are valid for the mechanical oscillator with:
k = 2ks
and
x = x1 − x0 .
III. APPARATUS
13
II.B Driven Oscillations
The case of a driven, damped harmonic oscillator is described in detail in [French]. The
equation of motion now includes a harmonic driving force, F0 cos ωt:
dx
F0
d2 x
+ ω02 x =
cos ωt
+γ
2
dt
dt
m
(3.5)
The steady state solution to this equation is: (see Eqs.(4-11) and (4-12)
x = A(ω) cos(ωt − δ)
(3.6)
F0 /m
A(ω) = (3.7)
2
(ω0 − ω 2 )2 + (γω)2
γω
.
(3.8)
tan δ(ω) = 2
ω0 − ω 2
Let us now examine the case at hand. Point D is now moving back and forth, driving
the oscillation. (See Figure 3.1). The force due to the first spring remains unchanged:
F1 = −ks (x1 − x0 )
The force due to the second spring is now altered by the relative position of point D
with respect to its midrange position. Let xd − xdo be the displacement of point D from
its midrange position. The force due to the second spring is now:
F2 = −ks [(x1 − x0 ) − (xd − xdo )]
For a harmonic drive with amplitude B, this relative position can be expressed as
B cos ωt:
F2 = −ks [(x1 − x0 ) − B cos ωt]
The total restoring force is now (still assigning x = x1 − x0 and k = 2ks )
Fr = −kx +
Eqns.(5)–(8) are therefore valid, with F0 =
kB
cos ωt
2
kB
2
=
mω02 B
:
2
x = A(ω) cos(ωt − δ),
ω02B/2
(3.9)
,
(ω02 − ω 2)2 + (γω)2
γω
tan δ(ω) = 2
.
ω0 − ω 2
The “Q” for the driven, damped oscillator is described in detail in pages 96-101 of
[French].
A(ω) =
14
EXPERIMENT 3. THE MECHANICAL OSCILLATOR
light beam
Amplitude Detector
Screen
microswitch
Spring 1
Driver
Spring 2
Figure 3.2. Experimental Apparatus
III. Apparatus
The apparatus consists of a mass which slides on a rod. The motion of the mass is
determined by the two springs attached to it. The end of one spring is held rigid, while
the other one is fastened to a driver which produces a linear harmonic motion of fixed
amplitude and variable frequency. The driver is powered by a variable speed motor. A
microswitch attached to the driver closes when the driver is at maximum amplitude and
is used to produce an electrical pulse at that time. The displacement from equilibrium
of the mass is determined by the amplitude detector which consists of a screen, light
source and photocell. The light source produces a thin beam of light about 3/4” high.
The screen, which is attached to the mass, has a triangular aperture so that the vertical
size of the light beam reaching the photocell is proportional to the position of the mass.
The amplified output of the photocell is measured with an oscilloscope. The oscilloscope
is also used to measure the time between pulses produced by the timing microswitch,
and thus the period of the driving force can be determined. Both the amplitude and
the timing pulse can be simultaneously displayed on the scope in order to determine the
phase relationship between the two.
12-16 V DC Light Source
Screen
Photocell
To Oscilloscope
Vertical Input A
Amplifier
Figure 3.3. Amplitude Detector.
1.5 MΩ
Timing Microswitch
0.47 MΩ
To Oscilloscope External Trigger
and Vertical Input B
Figure 3.4. Timing Pulse.
IV. PROCEDURE
15
IV. Procedure
1. Determine the amplitude of the driver. Rotate the drive shaft by hand and measure
the total range of travel. Since the drive motion is harmonic, the total range of
travel corresponds to travel from −B to +B. The drive amplitude B is therefore
half of the total range of travel.
2. Calibrate the amplitude detector, i.e. measure the photocell voltage as a function
of displacement of the mass. Set the oscilloscope voltage to line trigger, sweep
speed 5 ms/cm and use the oscilloscope as a D.C. voltmeter.
3. Free oscillation. Obtain a photograph of the amplitude vs. time for free oscillation.
• Set the scope to trigger internally, + slope, + level, vertical gain ×10, 0.5 cm/sec
sweep speed. Connect the output of the photocell to channel A.
• With the motor drive turned off, manually displace the mass towards the
driving system.
• Let the mass go. Adjust the horizontal and vertical position of the trace on
the scope, and adjust the vertical gain so that the damped sine wave appears.
Make sure that the trace is centered on the horizontal axis. Repeat this with
several more trials until you are sure that the scope trace is O.K.
• Photograph the trace:
Ask the teaching assistant for instructions on how to use the camera.
4. Driven oscillations.
• Switch back to normal sweep.
• Trigger the scope externally from the timing pulse: +level, + trigger slope.
Turn the motor on and observed the timing signals on channel B. It should
look (more or less) like this:
Figure 3.5.
• Switch vertical amplifiers to the “chopped” mode and observe both the timing
signal and the signal from the mass.
• Adjust the speed of the drive motor at resonance. This will be where the
amplitude of motion of the mass is large and the phase angle between the two
is 90◦ . Vary the speed above and below resonance. Observe the 0◦ and 180◦
phase angles at low and high speeds.
16
EXPERIMENT 3. THE MECHANICAL OSCILLATOR
Figure 3.6.
• Take photographs of the oscilloscope trace at speeds where the amplitude is
100%, 80%, 60%, 40%, and 20% of the maximum amplitude. Do this for
speeds above and below the resonance speed. Be sure to record the vertical
sensitivity (volts/cm) used in Channel A. Here is the sequence for picture
taking: With normal triggering, adjust the sweep speed and vertical position
for the desired picture. Close camera, switch to single sweep, with the ready
light off, open the shutter, press the reset button, close the shutter when the
sweep is over.
V. Results
V.A Free Oscillations
1. Determine the frequency of free oscillation, ω
2. From a semilog plot of amplitude versus time, determine the damping constant γ.
From this and ω, determine ω0 and Q. Be sure to state the uncertainties in your
experimental values.
V.B Driven Oscillations
1. At each frequency photographed, find the phase angle. Plot the results as a function
of frequency. From this graph, determine ω0 , γ, and Q. Using the values of ω0 and
γ found from the free oscillation photograph, calculate δ as a function of ω and
plot this on the same graph.
2. At each frequency photographed, find the amplitude of oscillation. Convert the
voltage readings to actual displacements. Plot the amplitude as a function of ω.
Find ωm (i.e. the frequency which results in maximum amplitude of oscillation; see
eq (4-15) in your book), and Q from this graph. From this graph, there are two
ways of finding Q. Using the values of ω0 and γ found from the free oscillation
data, calculate the expected amplitude as a function of ω and plot it on the same
graph as your data.
V. RESULTS
V.C
17
Error analysis
How well do the various values of ω0 and γ agree? Is there any evidence of systematic
error in this experiment? If so, suggest sources!
Experiment 4
Coupled Electrical Oscillators
I. Purpose
The purpose of this experiment is to:
1. Observe the transient behavior of a simple harmonic oscillator which undergoes
light damping. The frequency of oscillation and the decay constant of the oscillator
will be measured and compared to the theoretical value.
2. Observe the behavior of two simple harmonic oscillators which have been coupled
together. Their normal modes and a “beat” phenomenon will be observed. The
period of the observed motion will be measured and compared to theory.
II. Theory
Each electrical oscillator in this experiment consists of an inductor L connected to a
capacitor C, as shown in Figure 4.1a. Examples of other simple harmonic oscillators are
shown in Figures 4.1b and 4.1c:
L
C
k
m
(a)
(b)
(c)
Figure 4.1. Simple Harmonic Oscillators
A capacitor, C1 may be employed to couple the two oscillators together, as shown in
Figure 4.2a. Examples of other coupled oscillators are shown in Figures 4.2b and 4.2c:
The equations describing the behavior of these electrical oscillators will be summarized for the cases of free and coupled oscillation in the following sections.
18
II. THEORY
19
C1
L
C
C
L
k
k
m
(a)
m
k1
k
m
m
(b)
(c)
Figure 4.2. Coupled Harmonic Oscillators
II.A Free Oscillations
The differential equation describing the undamped LC oscillator of Fig. 4.1a is: (see Eq.
(4-30))
d2 q
1
L 2 + q=0
dt
C
A resistance should actually be included in the circuit diagram to represent the small
resistance which must always present in any real circuit. (Figure 4.3)
L
C
Rp
Figure 4.3. LC Oscillator
The resistance is denoted Rp since it is a small “parasitic” resistance due to the circuit
wires, as opposed to an actual resistor included in the circuit. The differential equation
will now be written: (see eq. (4-31))
1
d2 q Rp dq
+
q=0
+
2
dt
L dt LC
In analogy with a mechanical oscillator, (see Experiment #3, Eqs. (1)-(2)) this equation
may be rewritten as:
d2 q
dq
+ ω0 2 q = 0
+
γ
(4.1)
2
dt
dt
1
Rp
γ=
ω0 2 =
(4.2)
L
LC
Eq (1) describes the behavior of the charge q on the capacitor as a function of time. In
the laboratory, we will not measure the charge directly. Instead, we will measure the
voltage across the capacitor, Vc = Cq . For a small resistor, (i.e., a light damping force),
the solution to Eq (1) in terms of the voltage Vc is:
γ
Vc = V0 e− 2 t cos(ωt)
(4.3)
20
EXPERIMENT 4. COUPLED ELECTRICAL OSCILLATORS
R2
1
γ2
=
− p2
(4.4)
4
LC 4L
where V0 is the voltage across the capacitor at t = 0. The voltage therefore decays
exponentially with time and the frequency is slightly different from its undamped value.
ω 2 = ω02 −
II.B Coupled Oscillators
For the sake of mathematical simplicity, the resistance Rp in the oscillators will now
be ignored. Although the resistance has a very noticeable effect on the amplitude of
oscillation, it has only a small effect on the frequency. In this part of the experiment,
only the periods of the various observed motions will be calculated, and therefore the
resistance will be taken as zero. Two identical LC oscillators are capacitively coupled
together as shown in Figure 4.4.
C1
q3
i1
+
C
q1
L
i3
i2
q2
L
-
+
C
-
Figure 4.4. Coupled LC Oscillators
The equations describing this circuit are:
q1
d
= L (i1 − i3 )
C
dt
(4.5)
d
q2
= L (i2 + i3 )
C
dt
(4.6)
q1
q2
q3
,
=
−
C
C
C1
so
i3 =
C1
(i2 − i1 )
C
(4.7)
using the equations in (7) above to eliminate q3 and i3 from Eqs. (5) and (6), two coupled
equations describing i1 and i2 are found:
d
C1
C1
q1
=L
i2
i1 1 +
−
C
dt
C
C
(4.8)
II. THEORY
21
d
C1
C1
q2
=L
i1
i2 1 +
−
(4.9)
C
dt
C
C
Note the each equation is a function of two variables, namely i1 and i2 . Taking the sum
and difference of Eqs. (8) and (9), two new equations are found:
(q1 + q2 )
d
1
= L (i1 + i2 )
C
dt
(4.10)
L
1
d
= (C + 2C1 ) (i1 − i2 )
(4.11)
C
C
dt
These equations have the useful property that they can be written as a function of a
single variable with the substitutions:
(q1 − q2 )
q+ = q1 + q2
q− = q1 − q2
Making these substitutions lead to the pair of equations:
d2
1
q+ +
q+ = 0
2
dt
LC
(4.12)
0
q+ = q+
sin ω+ t
(4.13)
1
d2
q− = 0
q− +
2
dt
L(C + 2C1 )
(4.14)
0
q− = q−
sin ω− t
(4.15)
ω− =
1
L(C + 2C1 )
(4.16)
Thus the variables q+ and q− each behave as a simple harmonic oscillator with frequencies
of oscillation:
1
1
√
and 2π LC
2π L(C + 2C1 )
The variables q+ and q− are called the normal coordinates for the system. Eqs. (10)
and (11) each describe situations where the coupled oscillators act as a single harmonic
oscillator. These situations are called the “normal modes” of the system. In general, the
system motion can be more complicated than one of the normal modes. These modes
represent rather simple, but special cases of behavior, since the system can always be
described as a linear combination of the two modes, i.e., q = Aq+ + Bq− . The constants
A and B are determined by the initial conditions, i.e. the values of q1 , i1 , q2 , and i2 at
t = 0. In this experiment three cases will be observed: For all three cases the initial
currents will be zero, but the initial voltages, corresponding to the initial values of q1
and q2 will be:
q1 = q2 at t = 0. This corresponds to the normal mode q+ and is called the symmetric
mode since each oscillator is oscillating in phase at the same frequency.
22
EXPERIMENT 4. COUPLED ELECTRICAL OSCILLATORS
q1 = −q2 at t = 0. This is the normal mode q− and is called the antisymmetric mode
since each oscillator is oscillating at 180◦ out of phase.
q1 = q0 , q2 = 0 at t = 0. In this case, one of the oscillators is given an initial amplitude
and the other is not. Here the subsequent behavior of the system is not described by
either of the normal modes alone. The behavior is more complex and is described
by a sum of equal amplitudes of each mode:
q = q0 (sin ω+ t + sin ω− t)
Since the two normal mode frequencies are not equal, the resulting sum is not
a single harmonic function. For weak coupling, (C1 < C), ω+ ≈ ω− and the
two normal modes are close in frequency. When two functions of nearly equal
frequency and identical amplitudes are added, the resulting behavior has a beat
phenomena. This can be easily seen by applying trigonometric identities to the
previous expression for q:
q = 2q0 sin
ω+ + ω−
t cos
2
ω+ − ω−
t
2
If ω+ ≈ ω− ≈ ω0 , the behavior can be thought of as one sine function of frequency
nearly equal to ω0 , but whose amplitude slowly fluctuates between 0 and 2q0 at
a frequency of (ω+ − ω− )/2. The beat frequency (ω+ − ω− ) is double the naïve
frequency (ω+ − ω− )/2 as it corresponds to the frequency of successive maximum
values. The behavior can be thought of as a pair of oscillators each oscillating at
the average of the normal mode frequencies and the energy of oscillation slowly
going back and forth between them with the beat frequency.
III. Apparatus
Electrical energy is supplied to the oscillator by means of a square wave which is produced
by a timer chip (#555) which is powered by a 5 Volt DC supply. Each transition of the
square wave gives an initial amplitude to the oscillator. The period of the square wave
is chosen so that there is sufficient time between transitions (Figure 4.5) for the induced
oscillations to die down before the next transition occurs. The oscillations are observed
by means of an oscilloscope connected across the capacitors. The apparatus for this
experiment consists of a 5 volt power supply, a circuit for supplying the appropriate
waveform to the oscillators, a capacitance box, two large wirewound inductors, a dual
trace oscilloscope, and an oscilloscope camera. The circuit for supplying the appropriate
waveforms, one of the oscillator capacitors, and the coupling capacitor are contained in
a small grey box. The circuit diagram to this box and its connections are shown in
Figure 4.6.
A capacitance box is used to supply the capacitor for the second oscillator and instructions are given below for adjusting its value so that the two LC oscillators are identical.
IV. PROCEDURE
23
V
t
(a) Square Wave Function
V
e
-
R
t
2L
t
(b) Transient Response to a Square Wave Function
Figure 4.5.
There are several switches on the grey metal box. Switch 1 turns on and off the coupling
between the two oscillators. Switch 2 determines whether or not oscillator B will be
given an initial amplitude which is in phase (normal ) or out of phase (inverted ) with
oscillator A. Switch 2 may also be set to off, so that no initial amplitude is given to
oscillator B.
IV. Procedure
1. Connect the 5 Volt DC power supply to the connectors ground and +5V. This
supplies power to three chips within the gray box. The first chip produces a square
wave output. The second and third chips amplify and invert the square wave to
produce appropriate waveforms for the oscillator circuits.
2. Turn on the oscilloscope. Connect the trigger connection on the grey box to the
external trigger input via a t-connector and continue with a cable to the vertical
mode. Adjust the triggering stability control for maximum sensitivity by first
setting the trigger stability knob just below the point where the sweep (trace)
triggers freely. If your vertical and sweep controls are set properly you can now
adjust the trigger level for a stable trace.
3. Set the coupling switch to off and the second switch to normal. Check that the
inductors and the capacitor box are properly connected to the gray box. Set the
oscilloscope for internal triggering and adjust the trigger to obtain a stable sweep.
Set the scope for dual trace operation and connect the oscillators to the vertical
inputs of the scope.
24
EXPERIMENT 4. COUPLED ELECTRICAL OSCILLATORS
Inductors
To Oscilloscope Inputs
Osc A
Osc B
Cx
I-A
Square Wave Input
Gray Box
Cap Box
L
L
Output B
Output A
C1
S1
C
Cx
S2
normal
Square Wave
Input
off
S3
Buffer
Inverter
inverted
Figure 4.6.
4. Adjust the capacitor box so that both oscillators have the same frequency. After
this adjustment is made, photograph the trace and record on the back of the photo
the sweep speed which was used, i.e., how much time corresponds to one division
on your photograph.
5. Now turn on the coupling. Does the waveform change? Why or why not? Photograph the waveforms for the cases of the second oscillator is being driven normally,
inverted, and not driven (grounded). These three cases correpond to those discussed at the end of Section II.
6. Using the General Radio Bridge, measure the value of L and Rp of the inductors.The
GR bridge is described at the end of this experiment. Use the capacitance meter
to measure the capacitance box at the setting used in the experiment
V. RESULTS
25
V. Results
1. From the photographs, determine the frequency of the uncoupled oscillators and
from a semi-log plot of uncoupled amplitude versus time, find the decay constant
γ.
2. Using your data, determine ω = (ω+ + ω− )/2 and the beat frequency. Compare
these with the theoretical values.
VI. Using the GR Bridge
The General Radio bridge is a general purpose impedance measuring device. It operates
on the same principle as a Wheatstone bridge, that is, it is a nulling device. An unknown
impedance is placed in one leg of a bridge circuit and an adjustable known impedance
is placed in the other leg. The adjustable impedance is adjusted until the impedances
in each leg are as equal as possible. The difference in the impedances is indicated by a
current flow between the legs.
The overall procedure is then to adjust the known impedance until the meter reads
as close to zero as possible, increase the meter sensitivity, and iterate the process.
VI.A Detailed Instructions
This section is appropriate for measuring an impedance best represented as a pure inductance, L, in parallel with a pure resistor, R.
1. Turn the generator switch to bat check. The meter should point within the
bat section; otherwise replace the battery.
2. Turn the generator switch to ac internal 1 kHz. This will turn on an internal
sine wave generator.
3. Turn the parameter switch to Lp , (inductor in parallel with resistor).
4. Connect the unknown inductor between the high and low terminals.
5. Turn the orthonull switch to out.
6. Turn the osc level fully clockwise.
7. Turn the dq dial near 5 on the high q scale.
8. Turn the cgrl dial near 11.
9. Adjust the det sens for about 6 divisions of deflection.
10. Turn the multiplier switch until a minimum meter reading is obtained.
26
EXPERIMENT 4. COUPLED ELECTRICAL OSCILLATORS
11. Now alternately adjust the cgrl and dq dials and the det sens dial to achieve
the best possible null reading, i.e., the one with the smallest deflection from zero
for the highest possible det sens setting.
It is now possible to read off the inductance and the Q of the inductor. The inductance
will be the product of the cgrl dial and the multiplier setting. The Q is the reading
of the dq dial divided by the generator setting (typically 1 kHz).
Please remember to turn the generator switch to off.
Experiment 5
Pulses on a Coaxial Cable
I. Purpose
• To measure the pulse amplitude and time intervals using the external triggering
feature of an oscilloscope.
• To investigate the behavior of “fast” pulses and to measure the velocity of propagation and the effect of resistive terminations on coaxial cables.
II. Background Information
II.A Electrical Pulses
In this experiment rectangular pulses will be used. The pulse amplitude, duration (often
called width), and rise time are illustrated in the figure below:
Voltage
V0
0.5 V0
duration
Amplitude
rise time
Time
Figure 5.1.
When the pulse rise times and durations are comparable to the time it takes for the
pulse to travel from point to point in a system, special problems arise due to the effects
of the inductance and capacitance of the conductors. These difficulties arise when the
pulse shape has time intervals of 100 ns or less. (1 ns = 10−9 sec; speed of light ≈ 1 ft/ns)
27
28
EXPERIMENT 5. PULSES ON A COAXIAL CABLE
Electrical pulses in this time domain are referred to as fast pulses and the techniques
and equipment needed to handle them are often referred to as fast electronics. The
major difficulty in working with fast pulses is that the effect of lead inductance and
stray capacitance (capacitance between leads and nearby large metallic surfaces such as
chassies, etc.) can distort the pulse shapes while for slow pulses (times below 100 ns)
these effects are negligible. To overcome such problems, transmission lines are used to
transport fast signals over large distances (greater than a foot or so) without distortion.
II.B Transmission of Fast Pulses
A transmission line is used to transmit fast pulses or high frequency signals over distances
of a foot or longer, without distortion. A transmission line is a pair of conductors arranged
so that their geometrical cross-section remains constant along the entire length of the
conductors. Common types of transmission lines are twin lines, coaxial cables, and strip
lines. Their cross sections are shown below.
Dielectric
Conductor
Twin Line
Coax
Strip Line
Figure 5.2.
The electrical properties of the transmission lines depend on their resistance, inductance, and capacitance. The ohmic resistance of transmission lines is usually negligible.
The inductance arises from the self inductance of the conductors and the capacitance
from the capacitance between the two conductors. The inductance and capacitance are
distributed uniformly along the length of the line and are described by giving the inductance and capacitance per unit length. The equivalent circuit of a transmission line is
thus:
L
input
L
C
L
C
output
Figure 5.3.
Clearly, for an input voltage that does not vary with time (dc), the input and output
voltages will be equal and no current will flow. However, if the input voltage varies with
time, the effect of inductance and capacitance must be taken into account. This can be
II. BACKGROUND INFORMATION
29
done in a straightforward manner by writing down the differential equations obeyed by
the voltage and currents in this system. When this is done a rather simple and elegant
result is obtained. It is found that the time varying voltage and currents obey the wave
equations. Thus the electrical energy is transmitted down a transmission line in the
same way as sound waves travel through air or vibrations transmitted
on a string. The
√
velocity of propagation of electrical energy is found to be 1/ LC where L and C are the
inductance and capacitance per unit length. It is also found that the voltage and current
in the cable are in phase and that their ratio is given by L/C. This value is called
the characteristic impedance of the line. Recall that the definition of a resistance is a
circuit element or device for which the ratio of voltage across it to the current through
it is constant and in phase. Thus, for a time varying signal, the equivalent circuit for an
infinitely long transmission line is just a resistor whose value is that of the characteristic
impedance of the line.
A transmission line of constant cross section, i.e. constant characteristic impedance,
will in principle, transmit a pulse over an infinite length without distortion. In practice
distortion is introduced by the dielectric used to separate the conductors. In high quality
cable, an air core is utilized as the dielectric, and very little material is used to support
the center conductor.
Distortions are also introduced if the characteristic impedance of the line is changed,
for example, by connecting together two cables of differing characteristic impedances.
The optical/acoustical analogue of this situation is having the wave medium change index
of refraction/density respectively. In all of these cases the effect is the same, i.e., partial
transmission and some reflection occur at the boundary. The reflected wave can then
travel back down the line to the source (beginning of the line) and can be reflected again.
Multiple reflections can then be set up on the transmission line. If the times between
successive reflections are comparable to the pulse duration, then serious distortion of the
pulse can take place. On the other hand, if these reflections take place in times small
compared to the rise time of the pulse they will have little effect on the pulse shape.
II.C Behavior of Pulses at the end of the transmission line
In this experiment only the simplest case, which is the one most frequently encountered
in practice, will be considered. This occurs when the output end of a transmission line
is connected across a resistance. This may be a simple resistor, or the equivalent input
resistance of the device to which the end of the transmission line is connected. The effect
of the terminating resistance is to absorb part of the incident energy and reflect part of
it back down the line. The fraction of the incident pulse amplitude reflected depends on
the resistance of the termination compared to the characteristic impedance of the line.
The reflection coefficient is defined as the ratio of the reflected pulse amplitude to that
of the incident amplitude. Note that the the pulse that is present at the termination
of a cable is the sum of the incident plus reflected pulses. Also note that the reflected
wave is transmitted back down the cable where it eventually reaches the other end; it is
30
EXPERIMENT 5. PULSES ON A COAXIAL CABLE
reflected once again and multiple reflections can result.
Consider a coaxial cable terminated with a resistance R. There are three limiting
cases, illustrated in figure 5.4.
t = 0 sec
t = 75 nsec
t = 50 nsec
Open Circuit
Termination
zero
amplitude
Closed Circuit
Termination
R0
Matched
Termination
Figure 5.4.
At t = 0, a pulse is started down a cable whose delay time is 50 ns. The pulse is
shown at t = 50 ns when it has reached the end of the cable, and at t = 75 ns, when the
reflected pulse traveling back along the cable has reached the middle.
R = ∞, open circuit termination. The reflected pulse is of the same amplitude and
sign of the incident pulse. The pulse amplitude at the termination is twice the size
of the incident pulse.
R = 0, short circuit termination. The reflected pulse is of same amplitude but of
opposite sign of the incident pulse. The pulse amplitude at the termination is zero.
R = R0 , matched termination. R0 is the characteristic impedance given by R0 =
L/C. This resistance is the ratio of the voltage to the current at any point
in the cable where a pulse is present. When a cable is terminated in its characteristic impedance, it can be thought of as acting, from the point of view of the sending
end, as a cable of infinite length since there is no pulse reflected back. The incident
pulse is completely absorbed by the termination. This is an important case: when
using coaxial cable to transport signal from one place to another, and no reflections
can be tolerated, the far end of the cable must be properly terminated.
The general case can be described by defining a reflection coefficient. Rref , the ratio
of the reflected pulse to the incident pulse, is given by
Rref =
R − R0
.
R + R0
The reflection coefficient for the three special cases above are, from the formula, ±1,
0.
III. APPLICATION OF COAXIAL CABLES AND TRANSMISSION LINES
Type
RG 58 C/U
RG 59 B/U
RG 62/U
RG 114
β
65.9
64.9
84
85
v −1 ns/ft
1.5
1.5
1.2
1.2
C (pf/ft)
28.5
20.5
13.5
6.5
31
R0 (Ω)
50
75
93
185
Table 5.1. Characteristics of Some Common Coaxial Cable Types
III. Application of Coaxial Cables and Transmission Lines
1. Coaxial cables are used to transport signals from one place to another. Because
of their transmission line behavior, they can transmit pulses over long distances
without distortion. A properly terminated transmission line behaves as though it
were a resistor of value R0 , and the capacitance and inductance of the cable have
no effect. This is important when high frequency signals are being transported.
2. Delay lines. Delay lines are used in oscilloscopes, distributed amplifiers, pulse
coders and decoders, precise time measurements, radar, television, and in digital
computers.
3. Circuit elements. Very often in circuits operating with short rectangular pulses,
transmission lines can be used as elements to produce or shape pulses.
IV. Experimental Procedure
1. In the first part of the experiment, the delay time of a known length of cable will
be obtained directly by measuring the actual time taken for a pulse to travel from
one end of the cable to the other. To do this, advantage is taken of the fact that
the time the oscilloscope electron beam takes to travel across the screen depends on
the time elapsed between the starting of the sweep and the arrival of the pulse at
the vertical input of the oscilloscope. There are two methods for starting the sweep
(“triggering the sweep”). The internal triggering mode and the external triggering
mode. In the internal triggering mode, the sweep is started by the input signal
presented to the vertical amplifiers. In the external triggering mode, the sweep is
started by a separate signal which must be supplied to the external trigger input of
the oscilloscope. When used in this mode, the starting of the sweep is then locked
in time to the external trigger signal. In this mode, the position of a pulse on the
sweep will depend on the time relationship between the external trigger signal and
the pulse under observation.
We will use a dual-trace scope and select one channel to perform the triggering. We
will then be able to determine the difference in timing by examining the relative
delay between the two traces.
32
EXPERIMENT 5. PULSES ON A COAXIAL CABLE
The arrangement necessary to measure the delay time of a 100-foot RG-58 cable is
shown in figure 5.5.
A
B
Output
Figure 5.5.
Using a short RG-58 cable, connect the output of your pulse generator to one arm
of a t connector on channel A of your scope. Use the 100-foot cable to connect the
other arm to a second t on channel B. Put a 50-Ω terminator on the other arm.
Set the scope to trigger on channel A and observer both channels simultaneously
by setting either chop or alternate mode.
Adjust the time base as appropriate until you can see the details of both pulses on
the scope.
Using the positioning controls, shift the traces around until the half amplitude
point of the leading edge of the channel A pulse is located on the first vertical line
of the screen’s graticule. Observe and record the time shift of the pulse on channel
V. RESULTS
33
B with respect to this. This is the delay time of the cable. Is it close to what you
expect? (RG 58 delay time: about 1.5 ns/ft.)
2. In this part, the effects of varying the termination of a cable will be observed. Set
up the equipment as in the previous section.
Observe and record the wave forms produced for values of the terminating resistance
on channel B of 1000, 90, and 50 Ω. Record the width, amplitude polarity and the
time between the pulses (measured from leading edge to leading edge).
V. Results
Your report should contain a tabulation of measurements and results for the velocity of
propagation and comparison with the nominal values given in this write-up.
From the data obtained in step (2), explain the amplitude of the various pulses
observed and the time between them. From your data calculate the reflection coefficients
for the different terminations and compare them with the calculated values.
Experiment 6
Speed of Light
Caution: This experiment employs a 0.5 mW laser.
Please read the safety caution in the front of this manual. You will be observing the laser through a light
concentrating device, so extra care is required. There is
also a mirror which rotates at very high speed. It is enclosed in a shield, but we don’t want to have to replace
it or you. Operate with care.
I. History
299820
299810
299800
299790
299780
299770
299760
299750
299740
1930
1940
1950
1960
1970
Figure 6.1. Speed of Light versus Year. Figure taken from the
Review of Particle Properties, 1982 edition. Data from E.R. Cohen,
Rockwell International Science Center.
This speed of light (c) is perhaps the fundamental constant. (The meter is now defined
in terms of the speed of light and the second, c ≡ 299 792 458 m/s, see B.W. Petley,
Nature 303 373 (1983)). Due to its extremely high velocity, c is very difficult to measure.
34
II. APPARATUS
35
Galileo flashed signaling lanterns on distant hilltops and concluded that if light had a
finite velocity it was faster than he could measure.
Fizeau, in 1849, determined the speed of light in his laboratory to be 3.15×108 m/sec.
He used a rotating cogwheel and a mirror to generate short flashes. This was modified
by Foucault who replaced the cogwheel with a rotating mirror, developing the method
we will use in this experiment.
In the 1880’s, Michelson and Morley measured c to a much higher accuracy than
previously possible, and Michelson continued refining the measurement, but as figure 6.1
shows, work has continued throughout the twentieth century.
II. Apparatus
Fm
Fixed Mirror
L2
Rm
Rotating Mirror
s
L1
s'
Laser
Microscope
Figure 6.2. Apparatus Overview
Examine the diagram of the apparatus (figure 6.2). The parallel beam of light from
the laser is focused to a point image at s by lens L1 . Lens L2 is positioned so that the
image point at s is reflected from the rotating mirror (Rm ) and is focused onto the fixed,
spherical mirror (Fm ). Fm reflects the image back along the same path to again focus
the image at s.
In order that the reflected point image can be viewed through the microscope, a beam
splitter is placed in the optical path, so a reflected image is also formed at point s .
Now suppose Rm is rotated slightly so the reflected beam strikes Fm at a different
point. Because of the spherical shape of Fm , the beam will still be reflected directly
back toward Rm . An image of the source point will still be formed at s and s . The only
significant difference is that the point of reflection on Fm has changed.
However, when Rm is rotated continuously at high speeds, the image is no longer
formed at s and s . This is because, with Rm rotating, a light pulse that travels from
Rm to Fm and back finds Rm at a different angle when it returns than when it was first
36
EXPERIMENT 6. SPEED OF LIGHT
reflected. As will be shown in the next section, by measuring the displacement of the
image, the rate of rotation of Rm , the distance from Rm to Fm , and the magnification of
L2 , we can determine the speed of light.
II.A Geometric Evaluation
We begin by determining how the point of reflection on Fm relates to the rotational angle
of Rm . In figure 6.3, S − S0 is the displacement of the point of reflection on Fm caused
by a rotation of Rm by an amount θ − θ0 = ∆θ. The size of the mirrors and Rm are not
to scale.
θ
S
∆θ θ0
S0
2∆θ
α ∆θ
α
α+∆θ
Laser
Rotating Mirror
Figure 6.3.
Angle α is the angle of incidence of the light from the laser when Rm is at angle θ.
Since the angle of incidence equals the angle of reflection, the angle between the incident
and reflected rays is just 2α. When Rm is rotated by ∆θ, the angle of incidence increases
to α + ∆θ, and the angle between the incident and reflected rays increases to 2(α + ∆θ).
The angle between the two reflected rays is therefore 2(α + ∆θ) − 2α = 2∆θ. If D is the
distance between Rm and Fm , then
S − S0 = D 2∆θ = 2D(θ − θ0 )
(6.1)
Suppose Rm is rotating, and a single short pulse of light from the laser strikes Rm
when it is at angle θ0 . The pulse will be reflected to point S0 on Fm . Then by the time
the pulse returns to Rm , Rm will have rotated to a new angle, say θ. With Rm at θ, a
point reflected from S would be focused at point the point reflected from S0 will not.
However, S is in the focal plane of lens L2 , a distance ∆S away from the central point
of focus, S0 . To determine where the point image reflected from point S will be focused,
it is convenient to remove the confusion of the rotating mirror and the beam splitter by
II. APPARATUS
37
looking at their virtual images, as shown in figure 6.4. The critical geometry is the same
as for the reflected images.
S0 ∆S
S
L2
s
s0
∆S
s' s'0
∆s'
D
B
A
Figure 6.4. Apparatus Overview
Thinking in terms of the virtual images, the problem becomes a simple application
of thin lens optics. An object of height ∆S in the “plane” of Fm will be focused at the
plane of point s with a height of − oi ∆S. Here i and o are the distances of the lens from
the image and object, respectively, and the minus sign corresponds to the inversion of
the image. As shown, reflection from the beam splitter forms a similar image of the same
height. Therefore, ignoring the minus sign since we are not concerned that the image is
inverted,
S − S0
∆s = s − s0 =
(6.2)
(D + B)/A
Combining equations 6.1 and 6.2, the displacement of the image point relates to the
initial and secondary positions of Rm by
s − s0 =
2DA(θ − θ0 )
D+B
(6.3)
θ − θ0 depends on the rotational rate of Rm and on the time it takes the light pulse
to travel back and forth between Rm and Fm , a distance of 2D. Then
θ − θ0 = 2Dω/c
(6.4)
where ω is the rotational rate of the mirror in radians/sec. (2D/c is the time it takes
the light to travel from Rm to Fm and back.)
Using equation 6.4 to replace θ − θ0 in equation 6.3 results in
s − s0 =
Then
c=
4AD 2 ω
.
c(D + B)
4AD 2 ω
.
(D + B)(s − s0 )
(6.5)
(6.6)
38
EXPERIMENT 6. SPEED OF LIGHT
Equation 6.6 was derived on the assumption that the image point is the result of a
single, short pulse of light from the laser. But looking back at equations 6.1-6.4, the
displacement of the image point depends only on the difference in the angular position
of Rm in the time it takes for the light to travel between the mirrors. The displacement
does not depend on the specific mirror angles for a given pulse.
If we think of the continuous laser beam as a series of infinitely small pulses, the image
due to each pulse will be displaced by the same amount. All these images displaced by
the same amount will, of course, result in a single image. By measuring the displacement
of this image, the rate of rotation of Rm , and the relevant distances between components,
the speed of light can be measured.
III. Equipment
1. High Speed Rotating Mirror Assembly
This assembly comes with its own power supply and digital display of rotation
speed. The mirror is one centimeter in diameter and is flat to within 1/4 wavelength. It is supported by high speed ball bearings, mounted in a protective housing,
and driven by a DC motor with a mylar belt. A plastic lock-screw is provided to
hold the mirror in place during the alignment procedure.
An optical detector and digital readout provide measurements of mirror rotation
speed to within 0.1% and 1 rev/sec. The digital readout and the controls for mirror
rotation are on the front panel of the power supply. Rotation is reversible and
the rate is continuously variable from 100 to 1,000 rotations/sec. In addition,
holding down the max rev/sec button will bring the rotation speed quickly to
its maximum value at approximately 1,500 revolutions/sec.
2. Measuring Microscope
The 90X microscope is mounted on a micrometer stage for precise measurements of
the image point displacement. Measurements are most easily made by visually centering the image point on the microscope cross-hairs. The micrometer graduations
then permit the displacement to be resolved to within 0.005 mm.
To focus the cross-hairs, slide the eyepiece up or down in the microscope. To focus
the microscope, loosen the lock-screw on the side of the mounting tube and slide
the microscope up or down within the tube. In locating the image point, it is
sometimes helpful to remove the microscope from the micrometer stage. This can
be done by loosening the lock-screw that projects diagonally from one corner of the
micrometer stage.
3. Fixed Mirror
The fixed mirror is a spherical mirror with a radius of curvature of 12.5 m. It is
mounted on a stand and has separate x and y alignment screws.
IV. PROCEDURE
39
Eyepiece
Micrometer
Stage
Focusing
Lock-Screw
Micrometer
Knob
Beam Splitter
Angle Adjustment
Loosen to
Remove Microscope
Figure 6.5. Measuring Microscope
4. Laser and Alignment Bench
Use the OS-9171 PASCOS helium-neon laser. It has an output power of 0.5 mW
in the TEM00 mode and is randomly polarized. The wavelength of HeNe lasers is
632.8 nm.
5. Optical Bench
Use the OS-9103 1.0 m long optical bench. It provides four leveling screws and a
raised edge on the back used for aligning components along the optical axis.
6. Lenses and Auxiliary Equipment
You’ll need convex lenses of 48 and 252 mm focal length. Two polarizers are required to regulate the strength of the laser. Three component carriers are required
to hold these. Two alignment jigs are used to align the laser beam on the center
of the rotating mirror.
IV. Procedure
IV.A Alignment
All component holders, the beam splitter assembly, and the metal base for the optical
mirror assembly should be mounted flush against the fence (the lip along the back edge)
of the optical bench. This will insure that all components are mounted at right angles
to the beam axis.
1. Place the optical bench on a flat, level surface. Adjust the optical bench leveling
screws so the bench is level and stable.
40
EXPERIMENT 6. SPEED OF LIGHT
Rotating Mirror
Assembly
Rm
MicroscopeMicrometer Stage
L2
L1 Polarizers
Laser
Laser
Alignment Bench
Optical Bench
17 cm 62.2 cm
81.0 cm
93.0 cm
Leveling Screw
Figure 6.6. Equipment on the Optical Bench
2. Place the laser, mounted on the laser alignment bench, end-to-end with the optical
bench, at the end corresponding to the 1-meter mark of the scale. Place the bench
couplers as shown in figure 6.6, but do not tighten them yet.
3. Mount the rotating mirror assembly on the opposite end of the bench. Be sure the
base of the rotating mirror assembly is flush against the fence of the optical bench
and align the front edge of the base with the 17 cm mark on the bench.
The laser must be aligned so the beam strikes the center of the rotating mirror
(Rm ). Two alignment jigs are provided for this purpose. Place one jig at each end
of the optical bench with the edges flush against the fence. When properly placed,
the holes in the jigs define a straight line that is parallel to the axis of the optical
bench.
Turn on the laser. Check that the beam attenuator on the top of the laser, directly
above the aperture, is fully open.
Adjust the position of the front of the laser so the beam passes directly through the
hole in the first jig. (Use the two front leveling screws to adjust the height. Adjust
the position of the laser on the alignment bench to adjust the lateral position.)
Then adjust the height and position of the rear of the laser so the beam passes
directly through the hole in the second jig.
To fix the laser in position with respect to the optical bench, tighten the screws on
the bench couplers. Recheck the alignment of the laser.
4. Align the rotating mirror– Rm must be aligned so its axis of rotation is vertical
and perpendicular to the laser beam. Rotate Rm so that the laser beam reflects
back toward the hole in the alignment jig. Be sure to use the silvered side of the
IV. PROCEDURE
41
mirror as the reflecting surface, and tighten the lock-screw on the rotating mirror
assembly just enough so Rm holds its position as you adjust its rotation.
If needed, use pieces of paper to shim between the rotating mirror assembly and
the optical bench so that the laser beam is reflected back through the holes in both
jigs. Note: The beam need not go through the hole in the second jig, but it must
strike the second jig at the same height as the hole. Remove both alignment jigs.
5. Mount the 48 mm focal point lens (L1 ) on the optical bench so the centerline of
the component carrier is aligned with the 93.0 cm mark on the scale. Place a piece
of paper in front of the window of the rotating mirror enclosure to see the beam.
Without moving the component carrier, slide L1 as needed on its component carrier
to center the beam on Rm . (Notice that L1 has spread the beam at the position of
Rm .)
6. Mount the 252 mm focal point lens (L2 ) on the optical bench so the center line of
the component carrier aligns with the 62.2 cm bark on the bench. As for L1 in the
previous step, adjust the position of L2 on the component carrier so the beam is
again centered on Rm .
7. Place the beam splitter and microscope assembly on the optical bench so the front
edge of the micrometer stage is aligned with the 81.0 cm mark on the bench. The
lever that adjusts the tilt of the beam splitter should be on the same side as the
scale of the bench. Position this lever so it points directly down.
Caution: Do not look through the microscope until the polarizers have been place between the laser and the beam splitter.
The beam splitter will slightly alter the position of the beam. Readjust L2 on the
component carrier so the beam is again centered on Rm .
8. Place the fixed mirror (Fm ) at a distance of from 2 to 15 meters from Rm . The angle
between the axis of the optical bench and the line between Rm and Fm should be
approximately 12◦ . (If is greater than 20◦ , the reflected beam will be blocked by the
rotating mirror enclosure.) Also be sure Fm is not on the same side of the optical
bench as the micrometer knob, so you will be able to make the measurements
without blocking the beam. (It may be necessary to place them on the same side
given the physical layout of the lab. Do the best you can.)
Position Rm so the laser beam is reflected back toward Fm . Place a piece of paper
in the beam path to locate the beam and walk the beam back toward Fm so the
beam strikes it in the center. (A piece of paper against the surface of the mirror
will make it easier to see the beam.)
9. With a piece of paper still against the surface of Fm , slide L2 back and forth along
the optical bench to focus the beam to the smallest possible point on Fm . Then
42
EXPERIMENT 6. SPEED OF LIGHT
adjust the two alignment screws on the back of Fm so the beam is reflected directly
back to the center of Rm . This step is best performed with two people: one adjusting
Fm , and one watching the beam position at Rm .
10. Place the polarizers (attached to either side of a single component holder) between
the laser and the beam splitter. Begin with the polarizers at right angles to each
other, then rotate one until the image in the microscope is bright enough to view
comfortably.
11. Bring the cross-hairs of the microscope into focus by sliding the microscope eyepiece
in and out. Focus the microscope by loosening the lock-screw and sliding the scope
up and down. If the apparatus is properly aligned, you will see the point image
through the microscope.
If you do not see the point image, vary the tilt of the beam splitter slightly (no
more than a few degrees) and turn the micrometer knob to vary the transverse
position of the microscope until the image comes into view.
Note: In addition to the image point you may also see some extraneous
beam images resulting, for example, from reflection of the laser beam
from L1 . To be sure you are observing the right image point, place a
piece of paper between Rm and Fm while you watch the image in the
microscope. If the point does not disappear, it is not the correct image.
There may also be interference fringes visible in the microscope. These
fringes cause no difficulties as long as the point image is clearly visible.
However, the fringes and extraneous beam images can sometimes be removed without losing the point image. This is done by turning L1 or L2
slight askew.
If you still cannot find the point image, loosen the screw that attaches the microscope to the micrometer stage. Remove the microscope and place a piece of tissue
paper over the hole in the stage to locate the beam. Adjust the beam splitter angle
and the micrometer stage position to center the point image in the microscope hole.
If you still find no point image, try adjusting the longitudinal position of the micrometer stage along the optical bench. If this doesn’t work, recheck the alignment
starting from the first step.
When you find the point image, focus the microscope for the sharpest possible
image. Avoid sliding the microscope tube more than about a centimeter in its
mounting tube. If necessary, adjust the longitudinal position of the beam splitter
along the bench so the image point lies within the focal range of the microscope.
IV. PROCEDURE
43
IV.B Making the Measurement
The actual measurement is made by rotating the mirror at high speeds and using the
microscope and micrometer to measure the corresponding deflection of the beam image.
By rotating the mirror first in one direction, then in the opposite direction, the beam
deflection is doubled, and the effect of measurement error is reduced by half.
Important
Before turning on the motor, be sure the lock-screw for the mirror is completely loosened so the mirror rotates freely by hand.
Whenever the speed of the motor is accelerated, the red led on the front
panel of the motor control box will light. As the speed stabilizes, this light
should go off. If it does not, turn off the motor. Something is interfering
with the motor rotation. Check to be sure the lock-screw for Rm is fully
loosened.
Never run the motor with the max rev/sec button pushed for more than
one minute at a time.
1. With the apparatus aligned and the beam image in sharp focus, set the direction
switch on the rotating mirror supply to cw. Turn on the motor. Let the motor
warm up at about 300 revolutions/sec for at least three minutes. Then slowly
increase the speed of rotation and notice how the beam deflection increases.
Note: It may be necessary to adjust the polarizers, or even remove them,
in order to see the image point as Rm rotates. Be sure to replace them
if you intend to observe the image with the motor turned off.
2. Use the adjust knob to bring the rotational speed up to around 1,000 revolutions/sec, then push the max rev/sec button and hold it down. When the rotational speed stabilizes, rotate the micrometer knob on the microscope to align the
center of the beam image with the cross hair in the microscope that is perpendicular to the direction of deflection. Record the speed at which the mirror is rotating,
turn off the motor, and record the micrometer reading.
3. Reverse the direction of mirror rotation by switching the direction switch on the
poser supply to ccw. Again increase the motor rotational speed to around 1,000
revolutions/sec, then press the max rev/sec button. Hold the button down and
realign the microscope cross-hair with the center of the beam image. Record the
rate of mirror rotation, turn off the motor, and record the new micrometer reading.
Note: When reversing the direction of movement of the micrometer carriage, there will always be some movement of the micrometer knob before
the carriage responds (backlash). Though this source of error is small,
44
EXPERIMENT 6. SPEED OF LIGHT
it can be eliminated by always adjusting the initial position of the micrometer stage so that you approach the measurement position from the
same direction.
4. Equation 6.6, adjusted to fit the parameters just measured becomes
8πAD 2 (fcw + fccw )
(D + B)(scw − sccw )
(6.7)
Use this equation to calculate c. You can calculate A, by taking the distance between
L1 and L2 and subtracting the focal length of L1 . (See figure 6.7.)
Fm
Fixed Mirror
D
L2
S
L1
Laser
Rm
Rotating Mirror
B
S'
Microscope
A
fL1
Figure 6.7.
V. Notes on Accuracy
Precise alignment of the optical components and careful measurement are essential for an
accurate measurement. Beyond this, the main factor affecting accuracy is the distance
between the fixed and rotating mirrors.
The optimum distance between Rm and Fm is from 10 to 15 meters. Within this
range, accuracy within 1% is readily obtainable. If space is a problem, the distance may
be reduced to as little as 2 meters and an accuracy within 5% can still be achieved.
V. NOTES ON ACCURACY
45
In general, longer distances provide greater accuracy; Rm rotates farther as the light
travels between the mirrors, and the image deflection is correspondingly greater. Greater
deflections reduce the percentage of measurement error.
However, the optical components are designed for optimal focusing of the image point
at 12.5 meters (this is the radius of curvature of Fm .) Image focusing is not a significant
problem as long as the distance between the mirrors is within about 15 meters. At larger
distances, the intensity and focus of the image point begin to degrade, and measurement
and alignment are hampered.
Experiment 7
Diffraction and Interference
I. Purpose
To observe the intensity distribution of light emerging from a single and double slit.
II. Background Information
II.A Double Slit Interference
Consider plane waves of intensity I0 incident normally on two parallel slits of “zero”
width separated by distance d. The light is of wavelength λ and emerges from the slits
at an angle θ.
Incident Plane Wave
θ
Emerging Plane Wave
Figure 7.1.
The intensity pattern, according to simple wave theory, is shown in Figure 7.2.
Note the following features for this distribution:
• All points of maximum intensity are equal to 4I02 .
• The intensity is zero when
1
π,
α= n+
2
46
n = 0, 1, 2, 3, . . .
II. BACKGROUND INFORMATION
47
1
0.8
0.6
0.4
0.2
0
-2
-1.6
-1.2
-0.8
-0.4
0
0.4
0.8
1.2
1.6
2
COS(2*3.14159*X)**2
Figure 7.2. I(θ) = 4I0 cos2 α.
α=π
d
λ
sin θ.
α changes by π between successive minima, so
∆α = π =
πd
∆ sin θ
λ
Using the small angle approximation, ∆ sin θ = ∆θ, so
∆θ =
λ
d
Thus λ/d should give the angular spacing in radians between successive minima, and
for small angles, the spacing should be uniform. The maxima also should show the same
spacing.
II.B Single Slit Diffraction.
There is one obvious difficulty in the above theory, i.e., “zero” slit widths. With “zero”
slit widths the interference can be thought of as taking place between two point sources.
However, with slits of non-zero width, one has to not only account for the interference
from the wave fronts emerging from each slit, but the interference of portions of the
finite wave front in each slit. This latter effect is called diffraction. A simple wave theory
predicts that the intensity distribution of parallel rays emerging from a single slit is:
I (θ) = I0
sin β
β
2
where β =
πb sin θ
λ
(7.1)
48
EXPERIMENT 7. DIFFRACTION AND INTERFERENCE
I0 is the incident intensity, and b is the slit width. (see [Hallid] for details)
The intensity distribution is shown in Figure 7.3.
1
0.8
0.6
0.4
0.2
0
-2
-1.6
-1.2
-0.8
-0.4
0
0.4
0.8
1.2
1.6
2
(SIN(2*3.14159*X)/(2*3.14159*X))**2
Figure 7.3.
Note the following features:
• I (0) = I0
• Minima occur at β = mπ, m = 1, 2, 3, . . . The first minimum occurs at θ = ±λ/b.
After that the spacing is given by ∆θ = λ/b
• Because of the 1/β factor, the maxima are neither of equal intensity or equally
spaced. They approximately occur when sin β = 1 and β = (n + 1/2) π and n =
1, 2, 3, . . . The first maxima should be at an intensity of 9π4 2 of the central peak.
II.C Intensity Distribution from double slits of non-zero width
Simple wave theory predicts:
sin β
I (θ) = 4I0 cos α
β
2
2
(7.2)
which you should note is just the product of the “zero width” interference and single slit
diffraction intensity.
III. EXPERIMENTAL SETUP
49
Parallel
Laser
Light
d
θ
θ
Slit
f
Figure 7.4.
III. Experimental Setup
Laser light is incident on a slit. Parallel rays (for Fraunhofer diffraction) emerging from
the slit are focused on the light detector placed in the focal plane of the lens. (See Halliday
and Resnick to understand why the lens is necessary). Measure the light intensity as a
function of the distance d, of the light detector from the optic axis. (How is this distance
related to the angle θ of the emerging light?)
The parameters for the system are:
• λ = 632.8 nm (1 nm = 10−9 m)
• Slit Widths: (in mm) Single slits (0.02, 0.04); Double slits (0.04 widths, spaced
0.125, 0.25, 0.5 center to center.
• Lenses: 48 mm and 152 mm focal length available.
• Detector aperture: 0.1 mm (This is the approximate diameter of the fiber used to
transmit the signal to the photomultiplier.)
IV. Procedure
Read these instructions completely before starting the experiment.
The He-Ne laser is the most useful optical instrument to enter the student laboratory
in many years. However, it must be remembered that the laser is an optical instrument
capable of producing an intense light.
THIS IS NOT A TOY!
Although medical research has not produced any conclusive determination of what
laser power levels produce eye damage in human, present indications are that this threshold level is well below the 0.5 mW rating of typical lasers which we use. Nonetheless, one
should never look directly into the laser beam, nor should one look into the reflection of
a laser beam from a mirror. To prevent accidental eye contact with the laser beam, the
following precautions are suggested:
50
EXPERIMENT 7. DIFFRACTION AND INTERFERENCE
• Make lasers available for use only to persons who understand their proper handling.
• Turn the laser on only when necessary for the experiment.
• Keep the laser mounted on the optical bench at all times during operation of the
laser. Turn the laser OFF when moving it from one experiment to another.
IV.A Alignment of the Beam
1. It is frequently necessary to align the laser beam along the optical axis of the optical
bench. This may be performed using the alignment screws on the bench and shims
made from paper.
2. Attach the slide of a single slit on a component carrier and position the assembly
midway down the bench. (50 cm mark)
3. Attach the viewing screen (or a stiff piece of white paper) to a component carrier
and position it at the right end of the bench.
4. Adjust the slit position until the laser beam is incident and centered on the full
width of the slit. Use the 0.04 mm slit.
5. Observe the diffraction pattern on the screen. Adjust the screen position if necessary to obtain image clarity. This is an example of Fresnel diffraction.
6. Remove the slit from its carrier and place the 152 mm lens on a carrier positioned
at about the 50 cm point. Place the viewing screen at the end of the bench. Adjust
the lateral position of the lens so that its image is centered on the optic axis.
7. Replace the slit and measure and record the distance, D, between the first two
minima which are centered about the central bright spot, for at least six different
screen positions. Now calculate the diffraction angle, D/28, (8 is the lens to screen
distance) and compare it with the expected value of λ/b. (λ = 632 nm, b is the
slit width). Does the theoretical value fall within one standard deviation from the
average experimental value?
8. Adjust the photometer so that the central maximimum produced by the 0.1 mm
slit is centered on it. Position the linear translator so that the distance from the
152 mm lens mount and the aperture slide is 149 mm. Carefully align the total
system so that the diffraction pattern will be centered vertically over the complete
range of travel of the linear translator.
9. Place two component carriers with Polaroids between the laser and the slit. They
will be used to adjust the intensity of the laser to match the dynamic range of the
photometer. Even an “unpolarized” laser has a tendency to polarize when first
turned on, so let the laser warm up before making measurements.
IV. PROCEDURE
51
10. Turn on the photometer and adjust it so that it does not pin the meter at the
center of the diffraction pattern. Check the background by blocking the laser beam
and zero the meter. Readjust the sensitivity if necessary.
11. Manually scan through the diffraction pattern and note the maximum intensity. If
necessary, change the attenuator and sensitivity.
12. Turn on the chart recorder power. Attach the analog output of the photometer to
the vertical input of the chart recorder. Block the laser light to the photometer
and zero the chart recorder. Start both the translator and the chart recorder at
the same time. Record two diffraction patterns.
13. Replace the single slit with double slits of 0.4 mm width and 0.125 mm spacing.
Increase or decrease the attenuation if necessary and record two patterns.
14. Plot/tabulate the theoretical values of intensity versus position and compare them
with your measured data on the chart recorder.
Experiment 8
Polarization
I. Purpose
The purposes of this experiment are to observe and measure several polarization phenomena. Linear polarization produced by a dichroic substance (Polaroid filter) and by
reflection from a glass surface will be studied.
II. Background Information
II.A Linear Polarization
Diffraction and interference effects occur with any type of wave. However, since light is a
transverse electromagnetic wave, it exhibits one property not common to all waves. This
unique property is polarization. As a light wave travels through a medium, the electric
and magnetic fields oscillate in a plane perpendicular to the direction of travel. If all
light waves from a given source are such that their electric field vectors are parallel, the
light is said to be linearly polarized. The direction or axis of polarization is defined as
the direction of the electric field vector of the light wave. Consider an x-axis in the same
plane as the electric field vector and also perpendicular to the direction of the light beam.
Let θ be the angle between the x-axis and the electric field vector. (See Figure 8.1.)
E
θ
Plane Perpendicular
to
Light Beam
x
Direction of Light
Beam
Figure 8.1.
If E is the magnitude of the electric field vector, then its component along the x-axis
is E cos θ. Since the intensity of a light wave is proportional to E 2 , then the intensity
52
II. BACKGROUND INFORMATION
53
of a linearly polarized light beam measured at an angle θ to its axis of polarization is
I = I0 cos2 θ. I0 is the intensity of the light beam measured along its axis of polarization
and I is the intensity along another direction rotated by angle θ. This result is known as
Malus’ Law.
The light from ordinary sources is usually not linearly polarized. Most frequently, it
is randomly polarized (unpolarized). Unpolarized light has equal intensity for all possible
directions of polarization. However, it is possible for a light beam to have any degree of
polarization varying from the case of complete polarization to completely unpolarized.
Also, linear polarization is not the only form of polarization. Light can also be circularly
or elliptically polarized.
II.B Production of Linearly Polarized Light
Dichroic Material. (Polaroid Filter)
Some crystalline substances which exhibit a property called dichroism are useful in producing polarized light. A dichroic substance tends to absorb light to varying degrees
depending on the polarization of the incident beam. With proper engineering, the light
transmitted through such a substance is linearly polarized. Using such polarizers, the
transmitted light is linearly polarized with the electric vector parallel to the 0◦ − 180◦
axis of the polarizer.
Reflection.
Another way of producing linearly polarized light is by reflection. Electromagnetic wave
theory predicts that light is reflected to a greater or lesser degree depending on its
polarization relative to the surface of the reflecting medium. Hence, as Sir David Brewster
discovered in 1812, there is a predictable angle of incidence at which light reflected from
a medium is totally linearly polarized. At this angle of incidence, known as Brewster’s
angle, the incident and refracted rays are 90◦ apart. Using Snell’s Law, it is found that
n = tan θB where n is the index of refraction and θB is Brewster’s angle. At this angle,
the reflected light is completely polarized in the direction parallel to the surface of the
glass.
The polarization of the reflected rays for any angle of incidence is given by the Fresnel
Formulas:
IR1
R1 =
=
I01
IR2
=
R2 =
I02
sin (θ − φ)
sin (θ + φ)
tan (θ − φ)
tan (θ + φ)
2
(s-polarization)
2
(p-polarization)
and
sin φ = (sin θ) /n
(Snells Law),
54
EXPERIMENT 8. POLARIZATION
θ is the angle of incidence,
where φ is the angle of refraction,
n is the index of refraction.
I01 (IR1 ) is the intensity of the incident (reflected) light polarized along the direction
perpendicular to the plane containing the incident and reflected light rays. I02 (IR2 ) is
defined in a similar fashion. (See Figure 8.2.
I
incident
E
E
I reflected
θ
φ
I refracted
(a) Defining θ,φ
E
(b) s-polarization ( E out of page)
E
(c) p-polarization ( E parallel to page)
Figure 8.2.
III. Apparatus
You will need a polarized laser and a photodetector mounted on a rotating stage. (General information on laser use and safety is described in the write-up for Experiment 7:
“Diffraction and Interference.”) Make sure that you use the linearly polarized laser rather
than a laser with random polarization. Since you will then have a polarized light source,
you will not need to use an initial Polaroid to establish polarization.
IV. Procedure
1. Demonstrate Malus’ Law. To perform this measurement, the stage is used only
to mount the photometer sensor. Align sensor along the axis of the laser. Place
a Polaroid on a mounting backet between the laser and the sensor. Record and
plot the intensity as you change the angle of the transmission axis of the polarizer.
Compare the data with Malus’ Law by performing a fit to a cos2 θ distribution.
2. Demonstrate Brewster’s Law. Place a glass prism on the center section of the
rotating stage (see FIgure 8.3). Align the system so that the reflected beam falls
on the center of the photodetector sensor over the entire range of angles.
IV. PROCEDURE
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Rotating Stage
Prism
Photometer Sensor
Polarized Laser
Figure 8.3. Setup for Measuring Brewster’s Angle.
Reflect the laser beam off one face of the prism. Record the intensity of the reflected
beam and the angle of reflection, as a function of the incident angle. Try rotating
the laser 90◦ and repeating the measurement. Compare your results with theoretical
predictions for reflectance vs. angle of incidence for the two incident polarizations.
3. Multiple Polarizers. Orient a polarizer such that the recorded intensity is at a minimum. Now place a second polarizer between the laser and your original polarizer
and plot the intensity as a function of polarizer angle. Explain your result.
Bibliography
[French] A.P. French, Vibrations and Waves.
[Hallid] Halliday & Resnick, Fundamentals of Physics, 2nd Ed., pp. 741-745.
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