Download Document

Physics Laboratory Manual
PHYC 20080
Fields, Waves and Light
Name.................................................................................
Partner’s Name ................................................................
Demonstrator ...................................................................
Group ...............................................................................
Laboratory Time ...............................................................
1
2
Contents
Introduction
1
1
Waves and Resonances
7
2
Interference and Diffraction
13
3
Studying Sound using an Oscilloscope
23
Measurement of the Focal Lengths of Lenses and a
4
Determination of Brewster’s Angle
31
5
Magnetic Breaking
37
6
Determination of the Resistivity of a Metal Alloy using a
Wheatstone Bridge
43
3
Laboratory Schedule
Students will be assigned to one of a number of groups and must attend at a specified
time on Tuesday or Thursday afternoons. Check the notice-boards in physics to find
out which group you are in. In total you will perform six experiments during the
semester.
Room 127
Wk 1 (Sept 12-17)
Wk 3 (Sept 26-30)
Wk 5 (Oct 10-14)th
Wk 7 (Oct 24-28)
Wk 9 (Nov 7-11)
Room 131
Room 132
Magnetic
Breaking
Lenses
Resistivity
Waves
Interference
/Diffraction
Wk 11 (Nov21-25)
Sound
4
Introduction
Physics is an experimental science. The theory that is presented in lectures has its
origins in, and is validated by, experimental measurement.
The practical aspect of Physics is an integral part of the subject. The laboratory
practicals take place throughout the semester in parallel to the lectures. They serve a
number of purposes:
•
•
•
an opportunity, as a scientist, to test the theories presented in lectures;
a means to enrich and deepen understanding of physical concepts presented
in lectures;
the development of experimental techniques, in particular skills of data
analysis, the understanding of experimental uncertainty, and the development
of graphical visualisation of data.
During the laboratory practicals, students will be paired off with a partner with whom
they perform the experiment and collect the data. In order to make optimal use of your
time in the laboratory, please read the manual and familiarise yourself with the
experiment before the laboratory practical.
This manual includes blanks for entering most of your observations. Additional space is
included at the end of each experiment for other relevant information. All data,
observations and conclusions should be entered in this manual. It is advisable to use a
pencil when entering data in case of errors. Calculations are best performed in rough
before entering the salient details in the space provided. Graphs may be produced by
hand or electronically (details of simple computer packages will be provided) and should
be secured to this manual on one of the blank pages at the end of each experiment.
Your work will be collected and marked by your demonstrator and will contribute 23% of
your final mark.
Before leaving the laboratory, ensure that your lab demonstrator signs and dates the
data you have taken.
5
Summary on combining uncertainties
Suppose you calculate f which is some function of variables x and y. Both x and y have
associated uncertainties
δx
and
δ y respectively.
δf =
So long as x and y are uncorrelated,
What is the uncertainty
δf
on f ?
∂f
∂f
δx ⊕ δy.
∂x
∂y
This expression is true in general, but let us summarise the three most usual cases.
Case 1: f=Ax
You just multiply x by a constant A; then
δ f = Aδ x
Case 2: f=x+y or f=x-y
You add or subtract two independent measurements; then
δ f = δx ⊕δy
Just add the errors in quadrature!
(Short cut: If one error is at least twice as big as the other, you can ignore the smaller
error. So in the case that δ x
> 2δ y then δ f ≈ δ x )
Case 3: f=xy or f=x/y
δf
You multiply or divide two independent measurements; then
f
=
δx
x
⊕
δy
y
Just add the fractional errors in quadrature!
(Short cut: If one percentage error is at least twice as big as the other percentage error,
you can ignore the smaller. So when
δx
x
>
δy
y
then δ f
≈
f
δx )
x
An alternative way to proceed
An approximate, “quick and dirty” way to work out the errors is to move all the values to
their upper bounds and calculate your answer. Then move them all to their lower
bounds and calculate the answer. From this you know the likely spread in your answer.
6
Waves and Resonances
Introduction
This experiment produces waves in a stretched wire and looks at how the frequency of
vibration is related to the tension of the wire, its length and the mass per unit length.
The phenomenon of resonance is used in the investigation.
Using Newton’s laws, it can be shown1
that the velocity, v, of a symmetrical pulse
on a string, as shown on the right, is
related to the tension of the string, T, and
the mass per unit length, µ, by
v= T
µ
(Eq. 1)
Thus the velocity of a wave along a string depends only on the characteristics of the
string and not on the frequency of the wave. The frequency of the wave is fixed entirely
by whatever generates the wave. The w
wavelength
avelength of the wave is then fixed by the
familiar relationship:
v = fλ
(Eq.2)
Example: A string on a bass guitar is 1m long and held under a tension of 100N. If the
string has a mass of 10g, what is the velocity of a wave on the string?
stri
1
The portion of the wire
a net vertical force of
∆l experiences a tension on each end.
2T sin θ ≈ 2Tθ = T∆ l R acts.
The horizontal components cancel and
Newton says this force = mass times
µ
acceleration. The mass is ∆l . Looking at the system from a frame where the pulse is at rest (and the
wire moves with velocity v),
), the portion of wire moves along the indicated circle and the centripetal
acceleration is
v2 R .
Thus T∆ l
R = ( µ∆l )(v 2 R ) and the result follows.
7
If a wave with a frequency of 80Hz is sent into the string, what will its wavelength be?
If a wave with a frequency of 50Hz is sent into the string, what will its wavelength be?
A resonant frequency is a natural frequency of vibration determined by the physical
parameters of the vibrating object. There are a number of resonant frequencies for a
string which are the multiples of its length that allow standing waves to be formed. Thus
λresonant = 2l / n .
where n=1,2,3...
What is the lowest resonant frequency (the fundamental) of the bass guitar string
referred to above?
What happens if a wave with a frequency which is the same as the resonant frequency
enters the string?
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
8
What happens if a wave with a frequency which is different to the resonant frequency
enters the string?
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
Experimental Procedure
In this experiment we will attempt to confirm Eq.1 above by sending waves of different
frequency into the wire and identifying the resonant frequency.
The apparatus is shown above and consists of a stretched string held under tension by
a hanging weight. In this configuration, the tension equals the force exerted by gravity.
The wire passes over a bridge which defines a node thus changing the length in which a
wave can resonate. A small horseshoe magnet should be placed half-way between the
bridge and the pulley. An AC generator is connected to either side of the wire and can
send electrical signals of selected frequencies down the wire.
Choose eight different positions of the bridge (remember to move the magnet too), and
find the lowest frequency at which maximum vibrations (resonance) occurs. This can
be observed by placing a folded piece of paper on the wire and noting when it gets
thrown off by the vibrations. Record your data below.
9
Wire tension
(N)
Resonant Frequency
(Hz)
Length
(m)
Mass/unit
length
µ=
Eq.1&2 can be combined to give
f =1
T
λ
µ
.
When this is at the resonant frequency
f resonant = n
2l
T
µ
(Eq. 3)
Plot the resonant frequency against 1/l.
What should the slope be equal to algebraically and numerically?
What is the slope of a straight line fit to your data?
10
1/Length
(m-1)
Comment on whether you consider Eq.3 has been verified and whether a good
agreement exists between theory and experiment.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
Now place the bridge in a fixed position. Vary the tension in the wire by adding or
removing weights, and find the lowest frequency at which maximum vibrations occur.
Length (m)
Resonant Frequency
(Hz)
Frequency squared
(s-2)
Mass/unit
length
µ=
Plot the square of the resonant frequency against the tension.
What should the slope be equal to algebraically and numerically?
11
Wire tension
(N)
What is the slope of a straight line fit to your data?
Comment on whether you consider Eq.3 has been verified and whether a good
agreement exists between theory and experiment.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
If you have time, set up a resonant position and write down the frequency. Now
increase the frequency until you see resonance again, and write down this frequency.
What do you notice? With reference to Eq.3, explain what is happening.
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
12
Interference and Diffraction
Introduction
The aim of this experiment is to investigate the interference of light. Shining light
through a single slit or a double slit produces characteristic diffraction patterns whose
features depend on the wavelength of the light and the size of the slit.
When light passes through a slit, or around an obstacle, diffraction effects are produced
if the dimensions involved are of the same order of magnitude as the wavelength of the
light. If the diffraction
ion pattern is viewed on a screen a set of alternate bright and dark
fringes will be observed. The diffraction can be of two types (a) Fresnel, - where the
object to screen distance is comparable to the object size and (b) Fraunhofer, where the
light is parallel
rallel and the distances involved are large. This is the easiest to treat
mathematically.
Investigation 1: Single Slit Diffraction
The figure above shows the intensity pattern as a function of angle when a single slit of
width a is placed in the path of a beam of light. Three different slit widths are shown.
From left to right they correspond to a size 10 times larger than the wave length (λ) of
the light, 5 times larger, and the same size as the wavelength of the light respectively.
The width of the pattern depends on the width of the slit; as the slit width is reduced, the
pattern broadens.
13
A schematic of the experimental arrangement is shown above. It can be seen that the
minima of the patterns satisfy
(Eq. 1)
asinθ = mλ
where θ is the angle the fringe makes with the beam axis and m is the order number.
For the 1st minimum, m=1 and since θ is small, sinθ ≈ tanθ = Y / D where Y is the
distance from the central maximum to the first minimum and D is the distance from the
pattern to the slit. Thus
a = λ D/Y
(Eq. 2)
Apparatus
A laser is used as the light source since it provides
a source of monochromatic coherent light. The
light passes through the slit and onto a screen.
Explain why we use
monochromatic source.
a
coherent
and
_______________________________________
_______________________________________
_______________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
14
Procedure
Align the laser beam so that it passes through the
slit and strikes the viewing screen. Make the
distance, D, from the slit to the screen as large as
possible in order to maximise the size of the image
and reduce your measurement error.
The diffraction pattern should look something like
the figure here. Identify the central maximum and
the first order minima around it. The distance
between these two minima is 2Y.
Given that the wavelength of the light is 692.8nm,
you should work out the width of the slit.
1.
2.
3.
Record your data below and find a value (with an uncertainty) for the width of the
slit.
Is it possible to measure the slit width using the second order minima? If so,
what value do you get and how does it compare?
If time permits, repeat these measurements for a second slit.
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
15
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
16
You can use this technique to measure very small distances with high precision.
If an object which presents a rectangular aspect to the beam (such as a wire or human
hair) is placed in front of the laser, a diffraction pattern will be produced identical to that
produced by a rectangular aperture of the same dimensions.
Find the width of a human hair. Record your observations below.
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
17
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
18
Investigation 2: Double slit Diffraction (Young's slits)
If a double slit is placed in the path of the beam, a Young's slits interference pattern is
obtained, which is superimposed on the single slit diffraction pattern as shown in the
diagram above. If d is the separation of the slits, constructive interference (a bright
fringe) is obtained when
nλ = dsinθ
(Eq.3)
When the angle is small this can be written as nλ
λ=dθ
(Eq. 4)
Therefore the angular separation between adjacent maxima is
λ/d = y/D
(Eq. 5)
where y is the linear seperation between adjacent maxima.
Procedure
Replace the single slit by the double slit. Move the slit carrier sideways until the laser
light passes through one slit ,when you will get a single slit diffraction pattern on the
screen. As you continue to move the slit carrier, the laser light will pass through both
slits and the diffraction will break up into an interference pattern.
Use Eq.5 above to estimate the slit separation. To do this, you will need to measure the
separation between the adjacent maxima which are very close together. To increase
your precision therefore, measure a number of bright fringes and divide by the number
of separations. (Careful: 2 fringes delineate just one gap.) Record your observations
below and make an estimate of the slit separation. Don’t forget your experimental
uncertainty.
19
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
20
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
21
22
Studying Sound using an Oscilloscope
Introduction
Sound is a longitudinal wave that propagates in solids, liquids or gases. In this
experiment you will measure the speed of sound in air by first measuring the
frequency, f, using an oscilloscope
loscope and then using a combination of the resonance
tube and the oscilloscope to measure the wavelength, λ, of the sound. The product of
these two quantities gives the speed of sound in air, c,, as shown by the equation:
c = fλ
(Eq. 1)
As sound passes a point in air, the pressure rises and falls in an almost periodic
fashion. For this experiment we will assume that the pressure changes are periodic. The
sound is generated by a tuning fork. When the tuning fork is struck off the bench tthe
ends vibrate at a fixed frequency. As the ends vibrate, they alternately compress and
rarefy the air next to them, setting up a pressure wave in the air.
The pressure wave is picked up by the microphone which acts as a transducer,
changing the sound
ound signal into an electrical signal. This can be displayed on the
oscilloscope and the frequency directly measured.
23
The Oscilloscope
An oscilloscope is a device for looking at electronic signals that may change in time. It is
a bit like the zoom lens on a camera: you can change the magnification on both the time
and voltage axes and see the signal in more detail. But changing the magnification (or
gain, to give it its proper title) does not change the signal itself. The oscilloscope
consists of a cathode ray tube (a TV tube) and four main electronic circuits to control the
screen all housed in one unit. We will now look at how to use an oscilloscope; for
details of its internal workings see the appendix .
The X-axis on the oscilloscope screen
is the time axis; the Y-axis is a
voltage axis. The trace that appears
on the screen may be measured by
counting the number of divisions
between features e.g. peaks.
In this case there are 4 divisions
between peaks, and the oscilloscope
is set to 2 ms per div.
With reference to the image
here:
What is the period (T) of the wave?
What is the peak-to-peak voltage?
From the value you have obtained for the period of the wave, you can calculate a
frequency using the relationship f = 1/T. What is the frequency of the signal displayed in
the image above?
(Note: convert the period to seconds before
calculating he frequency).
24
The oscilloscope can display waveforms for two signals which are input on one of two
channels (CH1, CH2). You can alter what you see using the controls on the front panel.
1
2
3
Position Control
Input
AC-GND-DC
4,5
6
7,8
9
10
11
12
Volts/Div
Mode
Volts/Div
AC-GND-DC
Input
Position Control
Pilot Lamp
13
14
15
16
Power Switch
Intensity Control
Focus Control
Ext. Trig
17
18
19
20
21
22
23
Source
Sync
Level
Pull Auto Switch
Position
Sweep Time/Div
Cal 1Vpp
Controls vertical position of spot for CH1
Input jack for CH1
If set to AC, only the AC component of the signal is
displayed. If set to GND, gives the zero voltage level. If
set to DC gives AC signal + any DC offset.
Controls vertical attenuator for CH1
select whether CH1,CH2 or both are displayed
As for 4,5, but for CH2
As for 3, but for CH2
As for 2, but for CH2
As for 1 but for CH2
Light when power switched on. Note, after switching off,
light remains on for a few seconds.
Switches on the unit
Adjusts the brightness of the spot on the screen
Brings the trace into focus
Connect to external trigger source when the source switch
(17) is EXT
Trigger for sweep is internal (INT) or external (EXT)
Determines point on waveform where sweep starts
Sweep generated without a trigger
Controls horizontal position of trace
Horizontal sweep time selector.
Provides 1V square wave pulse for calibration
25
Measuring the frequency of the sound
Turn the calibration knobs on the oscilloscpe fully clockwise. This ensures that the
settings for voltage and time are as indicated on the adjustable knobs. Connect the
microphone to Ch.1 and set the mode switch (6) to Ch. 1. When the tuning fork is
struck and held close to the microphone a sine wave will be displayed on the
oscilloscope. This will slowly drop in amplitude over about 20s. Why is this?
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
Measure the period four times at one timebase setting (dial 22) and four times at a
second setting. Each time the four readings should be taken at the following places:
1) any peak to any other peak; 2) any trough to any other trough; 3) any rising edge
across the 0 V axis to any rising edge; 4) any falling edge to any falling edge.
The period is found by counting the number of divisions between any feature on the
wave, multiplying by the sweeptime/div setting on the oscilloscope to get the total time,
and dividing by the number of wave cycles between these points to get the time for one
cycle. Tabulate your data below and calculate the frequency of the sound.
Feature
No. of
Divs
Sweeptime/div.
Time for N wave
cycles (s)
Period,
T (s)
Peak – Peak
Trough – Trough
Rising @ 0 V
Falling @ 0 V
Best value for T (average of these 8 readings)
(standard deviation gives uncertainty)
±
±
Frequency, f = 1/T
26
Measuring the wavelength of the sound
We will use the fact that sound resonates in
tubes. In this case, the tube is closed at one
end. This forces a node to occur at the closed
end and an anti-node
node to occur at the open end.
The longest wave (the fundamental) that can
exist in a tube like this is shown here.
You can see that when this resonant wave is present in the tube, its wavelength is 4
times the
he length of the tube. When using the tuning fork as a sound source the
wavelength is fixed. So we need to adjust the length of the tube until the condition
λ = 4L
is met. You will be able to tell that this is the case as the amplitude of
of the sound will be
at a maximum due to the air in the tube resonating in addition to the vibrations around
the tuning fork. With the microphone close to the open end of the tube, sound the
tuning fork. Start with no air in the pipe and increase the length
length of the open pipe until
you have maximised the amplitude of the signal on the oscilloscope. You should also
hear the sound grow louder as you approach the resonant condition. At the position of
greatest amplitude, measure the length of the air column, L,, in the open tube.
What is the wavelength of the sound?
±
If you increase the length of the tube still further, can you find any other positions where
resonance occurs? If so, why is this? Can you use these positions to calculate the
wavelength? Try it, and record your observations.
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
27
Measuring the Speed of Sound
Use Eq. 1 to calculate a value (with its uncertainty) for the speed of sound in air.
Comment on how it agrees with the accepted value of 332 ms-1 at 0C at sea level.
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
Appendix :The Cathode Ray Tube (CRT):
Fig. 1 Schematic Diagram of Cathode Ray Tube
A CRT consists of an evacuated glass tube. A hot filament (1) emits electrons.
These are accelerated and focused by the electron gun (2). The accelerated electrons
(5) strike the fluorescent screen (6) with sufficient energy to make it fluoresce, creating
a spot of light. On their way to the screen the electrons pass through two pairs of metal
deflecting plates (3) and (4). A voltage variation applied to the X-plates (4) causes the
electrons to deflect in the X- direction, the deflection being directly proportional to the
voltage.
28
Circuitry:
The four main circuits are the time base generator, the synchronising amplifier, the
display unit and the vertical amplifier.
(1) Time base generator produces a saw-tooth waveform.
This potential difference is applied across the X-plates. During the rising part of the
saw-tooth the electron spot is swept across the screen at a constant speed. During the
“fly back” of the waveform the electron gun spot returns swiftly to the start again. The
frequency of the saw-tooth is variable and hence the speed at which the spot travels
across the screen is also variable. The time base generator therefore produces a time
scale in the horizontal or X-direction controlled by the Sweeptime / div knob.
(2) The Synchronisation Amplifier ensures that the time base sweep and input signal
to the Y plates commence at the same time.
(3) The display unit controls the size of the spot and the overall brightness of the spot
on the screen.
(4) The vertical amplifier controls (a) the degree of amplification of the signals being
fed to the Y-deflection plates of the tube and (b) the vertical position of the spot on
the screen.
The oscilloscope used in this experiment is a two beam oscilloscope i.e.,
there are two spots which can move across the screen either one at a time or both
together allowing two traces to be displayed simultaneously if required.
Free Running Spot:
If the saw-tooth waveform is applied continuously in “free run” or “auto” mode this
voltage waveform makes the spot scan continuously across the tube face.
Internal Triggering:
A single cycle of the saw-tooth waveform may be initiated by triggering the
oscilloscope. The signal trigger can be derived from any signal applied to the Y-input or
vertical amplifier by the ‘scopes internal triggering circuitry. This causes a single sweep
of the beam displaying the time variation of the Y input signal for the duration of this
sweep.
External Triggering:
Alternatively, the signal trigger can be derived from some external source applied
via the external trigger input. This initiates a single sweep irrespective of what is
applied to the Y amplifier. This is used in the second experiments, which involve the
use of an oscillator ( or signal generator).
29
30
Measurement of the Focal Lengths of Lenses
and a Determination of Brewster’s
Brewster’s Angle
This experiment investigates the behaviour of light crossing from one medium to
another. In the first part you will determine the focal lengths of a concave and convex
lens. In the second part you will see how the reflected and transmitted
transmitted rays depend on
the polarisation of the light and determine Brewster’s angle, the angle at which no
parallel light is reflected.
Part 1: Determining the focal length of Lenses and Mirrors.
The phenomenon of refraction (or bending) of light at the interface
interface between two
materials can be used to make optical components to form images. Parallel light hitting
a spherical material will either converge or diverge depending on whether the surface is
convex or concave - to remember which shape is which, recall the entrance to a cave.
Thus both a convex lens and a concave mirror are converging. For spherical lens and
mirrors the distance of the object from the lens, u,, and the distance of the image, v, are
related by
1 1 1
+ =
u v f
(Eq.1)
where f is a constant for a particular lens or mirror, called its focal length
length.
(1) To determine the focal length of a convex lens
Set up the apparatus as shown here with
the light source passing through the lens
forming an image on the screen. The
object is a cross
s drawn on the frosted
glass of the projector. Place the source
50cm from the lens and move the screen
until you obtain the sharpest image. Adjust
the equipment so that the object and
image are both at the same height i.e. the
optical axis is horizontal and
nd parallel to the
edge of the bench.
31
Make at least five different measurements of source and image distances and enter
them in the table below with their associated uncertainties.
For each pair of measurements, calculate the focal length and enter it in the last
column. Find the mean and standard deviation of these focal lengths which are roughly
equal to the best estimation and its uncertainty.
u (cm)
δ u (cm)
v (cm)
δ v (cm)
f (cm)
Mean of the separate focal length determinations
Standard deviation of these determinations
Graph 1/v against 1/u.
What should the slope and intercept of the graph be?
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
Using the graph, calculate the best value of the focal length of the lens with its
associated uncertainty.
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
32
____________________________________________________________________
(2) To determine the focal length of a concave lens
Substitute the concave lens for the convex lens. Why can’t you proceed as you did for
the convex lens?
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
To overcome this problem, use the concave lens together with the convex lens. This
combined system will have a focal length, f, given by
1
1 1
= +
(Eq. 2)
f
f1 f 2
where f1 and f2 are the focal lengths of the individual lens. Proceed as before and fill in
the table below.
u (cm)
δ u (cm)
v (cm)
δ v (cm)
f (cm)
Mean of the separate focal length determinations
Standard deviation of these determinations
Use Eq.2 to find the focal length of the concave lens (with its associated error.)
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
33
Part 2: Determination of Brewster’s Angle
Light is an electromagnetic wave so it consists of electric and magnetic fields which
have an interesting property in that they are always perpendicular to the direction in
which the light is travelling.
Usually the direction of the electric field is
orientated in many planes (each
perpendicular to the direction of motion.)
The light is said to be unpolarised.
Light is coming
towards you. Electric
field oscillates as
shown by the arrows in
the plane perpendicular
to you.
However, it is possible to restrict the field
to a single plane when the light is said to
be plane polarised.
θi
When light passes from one medium to
another, some of the light is transmitted
and some is reflected. The angles of
θr
incidence θ i and reflection θ r are equal
while the angle of incidence and
transmission are related by Snell’s law:
n1
n2
n2 sin θ i
=
n1 sin θ t
θt
(Eq. 3)
How much light is transmitted and how much is reflected depends on the polarisation of
the light. Considering light polarised in the plane parallel to and perpendicular to the
plane of incidence separately, the ratio of the intensity of the reflected to the intensity of
the incident light is given by
R parallel
tan 2 (θ i − θ t )
=
tan 2 (θ i + θ t )
and
R perpendicular
sin 2 (θ i − θ t )
=
sin 2 (θ i + θ t )
(Eq.4)
so that if R=0 no light is reflected while if R=1 all the light is reflected.
The apparatus consists of a table on which the object to be measured is placed, and
two arms that are free to move. One arm holds the a laser whose plane of polarisation
is either known or can be set by using a polaroid, the other holds a photoresistor
detector whose resistance decreases the greater the intensity of light falling on it. Thus
a voltage measured across the resistor will increase in proportional to the intensity of
light. The angles of incidence and transmission can be read from a scale on the table.
Set the plane of polarisation of the light perpendicular to the plane of incidence. Vary
the angle of incidence and measure the intensity of the reflected light. Record your data
in the first two columns below and graph the results. Repeat your measurements for
light that is polarised parallel to the plane of incidence.
34
Light polarised perpendicular to the plane of incidence
Angle of Incidence
Voltage measured
Calculated angle of
(degrees)
(V)
transmission
(degrees)
Theoretical
value for
Rperpendicular
Light polarised parallel to the plane of incidence
Angle of Incidence
Voltage measured
Calculated angle of
(degrees)
(V)
transmission
(degrees)
Theoretical
value for
Rperpendicular
35
Comment on the curves you have obtained and find a value for Brewster’s angle θB, the
angle of incidence at none of the parallel polarised light is reflected: Rparallel = 0.
Calculate also the refractive index, n, of the material, related to Brewster’s angle by
n = tanθ B
(Eq. 5)
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
If time permits, use Eq.3 to fill in the third column of the tables above (the refractive
index of air can be taken to be 1). Then use Eqs. 4 to fill in the fourth column.
Compare these theoretical values to your experimental values. If possible, overlay
them on the same graph. Comment on the agreement.
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
36
Magnetic braking by electromagnetic induction
Introduction
When a magnet is dropped through a metallic pipe, electromagnetic induction produces
eddy currents, which in turn produce magnetic fields retarding the fall of the m
magnet.
The present experiment (1) investigates and quantifies this phenomenon and (2) also
finds a value for the magnetic moment of the magnet.
Background Theory:
Normally, when an object is released from rest and allowed to fall, it accelerates
downwards
ds at a rate equal to g,, the gravitational acceleration. However, when a
magnet is released from rest and is allowed to fall through a metallic tube, forces other
than gravity are called into action. From Faraday's law a changing magnetic field
passing through
hrough a conducting loop induces electric currents, called eddy currents, in the
loop. From Lenz's law the action of these induced currents is such that the magnetic
field produced by the current opposes the change in the magnetic field that produces
the current.
Thus, the falling magnet produces a changing magnetic field which exerts a force on it.
This force, which proportional to the velocity of the magnet, acts upwards against the
weight of the magnet. The falling magnet quickly reaches a terminal velocity
velocity (when the
upward force becomes equal to the downward force, resulting in zero acceleration) and
the currents generated reach a steady state value. The magnet then falls through the
remainder of the pipe at a constant velocity.
Theory:
Consider a magnet travelling at constant velocity, v,, through the pipe. By calculating
the magnetic force generated by the eddy currents (which are induced by the falling
magnet) and equating this force with the weight of the magnet, a value for the terminal
velocity, v,, can be found. The terminal velocity is given by the equation
v = 22.756 m g R4 / σ d µ0 2mB2
Eq. (1)
The other quantities are the mass,
mass m,, of the magnet, the annular radius, R, of the tube
(= 7.1 mm), the tube wall thickness, d (= 0.8 mm); mB is the magnetic moment,
moment µ0 is
-7
-2
the magnetic permeability of air (=4π×10
(=4
NA ) and the conductivity of the metal tubing
is σ = 5.964×107Ω-1m-1.
37
Figure 1. Effects of dropping a magnet through a conducting tube.
The value of the velocity, v, can be measured in the experiment, while all the other
quantities are well defined apart from the magnetic moment, mB, of the magnet. For a
point on the axis of the magnet, the axial magnetic field is
Bz(z )= µ0 mB / 2 π z3
Eq. (2)
where z is the distance along the axis from the centre of the magnet.
Procedure
The experiment consists of two parts. The first measures the terminal velocity of the
magnet through the tube. The second measures the field strength of the magnet in
order to find a value for mB or µo mB .
Apparatus
Copper pipe and support, magnet, magnetic field probe, timer and ruler.
Figure 2. Apparatus for measuring velocity of falling magnet and its magnetic moment.
38
Part 1. Measure the terminal velocity of the magnet through the tube
The apparatus consists of a vertically suspended copper tube and a magnet.
Measure the height, h, of the copper tubing, and take about 30 measurements of the
time, t, taken for the magnet to travel the length of the tube and record in Table 1 below.
This will give an accurate measurement of the average velocity, vave, of the magnet
through the tube which will be approximately equal to the terminal velocity, vt. The
annular radius, R, of the tube is the average of the inner and outer radii of the tube while
the wall thickness, d, of the tubing is half the difference of these radii. The values of
these are given below.
Theoretical
Time through
Time outside
Time of plastic Measured
Velocity, vmeas Velocity, Vcalc .
tube
Tube
through tube
[ms-1]
[ms-1]
[s]
[s]
[s]
Use Eq. 1(see later)
Mean =
Mean =
Mean =
39
Length of tube [m] =
Part 2: Determine the magnetic moment (mB )of the magnet
To measure the dipole moment, mB, of the magnet, place it on the small adjustable
slide. Set the multimeter on the DC 2000mV range. Move it the furthest distance from
the magnetic field probe and use the variable resistor to zero the reading on the
multimeter . Slide the magnet closer to the magnetic probe in small steps and record
the distance from the probe to the centre of the magnet, z, and the multimeter reading in
the Table below. This is converted to field strength, Bz using the conversion factor
1mV = 1.25 10-5 Tesla. Repeat this for up to 10 positions.
Position. z
[m]
Meter reading.
[mV]
Bz
[Tesla]
1 / z3
[m-3]
10 readings
Mean = ?
Plot a graph of Bz against 1 / z3. From Eq. 2, the slope of this graph is equal to
µ0mB/2π so that mB or µo mB can be evaluated.
Slope of Bz against 1 / z3 : the value of the slope =
Use this space to calculate mB and µo mB
40
±
mB =
±
µo mB =
±
In measuring the mass of the magnet, to avoid a contribution from the magnetic
attraction between the magnet and the balance, the magnet should be placed on a
teflon cylinder of suitable height.
Use the value for mB you just calculated to find the velocity as predicted by Eq 1
vcalc = 22.756 m g R4 / σ d µ0 2mB2 =
±
Compare this value to the measured result (vmeas) in table 1.
Calculate, using the standard laws of motion:
1. the average and final velocities of the magnet if it were dropped from a height,
h , but not through a conducting tube.
v final = 2 gh
vav =
=
±
vinit + v final
=
2
±
2. the final kinetic energies (KE) of the magnet after falling through the height, h,
KE final =
1
2
mv 2final =
±
3. Use vmeas (from table 1) to find the KE after it emerges from the conducting
pipe
KEmeas =
1
2
2
mvmeas
=
±
41
Why are these energies different and where does the missing energy go?
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
42
Determination of the Resistivity of a Metal Alloy
using a Wheatstone Bridge
Introduction
This experiment uses a simple but clever piece of apparatus, called the Wheatstone
bridge, to measure an unknown resistance to high precision. The Wheatstone bridge is
a circuit set up in a delicate balance so that no current flows through a galvanometer
(which is a very sensitive current measuring device). Any deviations from zero are
readily detected, so when the wheatstone bridge is balanced, you know to high
precision that no current is flowing in the galvanometer and this leads to high precision
in determining the resistance of an unknown substance. Wheatstone bridge circuits are
often at the centre of many electronic measuring devices, remote sensing equipment
and switches.
The resistance, R,, of a substance is directly proportional
propo
to its length, L,
L and inversely
proportional to its cross-sectional
sectional area, A,, as well as depending on the material it is
made from. Thus,
R = ρL
A
(Eq.1)
where the resistivity, ρ, depends on the material.
If you double the length of a resistor what happens to its resistance?
If you double the radius of a cylindrical resistor, what happens to its resistance?
43
Wheatstone Bridge circuit
The Wheatstone bridge is shown here and has
two known resistances
R1 , R3 ,
one variable
resistor, R2 , and the unknown resistor R X .
When the meter is balanced, i.e. when the current
through the galvanometer
RX
R3
= R2
Rg is zero:
R1
(Eq. 2)
(We will prove this later)
Apparatus
The apparatus used to measure the
resistance of the alloy is shown on the
right and schematically below. Note the
similarity with the Wheatstone bridge
above.
Instead of measuring R1, R2 however, we
take advantage of the fact that the
resistance of a resistor is proportional to its
length and so
R2
R1
=
L2
L1 (Eq. 3)
RX
R3
L2
L1
Method
44
Connect the apparatus as shown above, taking particular care that the contacts of R3
and RX are tight. Make sure that the longest length of RX is used, i.e. make the contacts
with the shortest possible lengths at each end and ensure that the wire cannot short
itself. Use the shortest possible lengths of wire to connect the known resistors to the
bridge. Press the contact maker against the wire and find two points, which give
deflections of the galvanometer in opposite directions. Between these points locate
accurately the point of contact which gives no deflection. Measure L1 and L2, noting
that L2 is the length opposite RX and L1 is the length opposite R3.
Interchange R3 and RX and repeat the experiment. (Lengths L1 and L2 will also swap:
note that L2 is always the length opposite RX. If the values of L1 and L2 for the second
balance point differs by more than 1 cm from the original, consult the demonstrator
before proceeding. The discrepancy may be due to bad contacts on the meter bridge.
Record your values in the table below. (Don’t forget to include errors.)
R3
L1
L2
RX
Average value for the unknown resistance of the alloy
Repeat the measurements for different values of the fixed resistor, R3.
R3
L1
L2
RX
Combine all your measurements to get the best value for the resistance of the alloy.
45
Detach the sample wire and uncoil it carefully so as not to produce any kinks. Measure
the length making allowance for the short lengths used at each end which were not
included in the resistance measurement.
Measure the diameter by means of a micrometer screw at about six places uniformly
distributed along the wire.
Record you measurements below and use Eq. 1 to calculate the resistivity of the metal
alloy along with its error.
46
Finally, prove what we have assumed all along,
that
RX
R3
=
R2
R1
Remember, when balanced, no current flows in
the leg DB.
The solution is outlined in the seven steps below.
1)
2)
3)
4)
5)
What current flows along DB?
What can you say about VDB, the voltage difference between D and B?
What can you say about VAD compared to VAB? Write an equation to express this.
What can you say about VDC compared to VBC? Write an equation to express this.
Let IL be the current that flows along the left hand line A->D->C and IR the current
that flows along the right hand line A->B->C. (Remember nothing flows from D->B)
6) Express each of VAD , VAB ,VDC , VBC as a current times a resistance using Ohm’s
Law.
7) Substitute into the two equations in steps 3&4 and divide one equation by the other.
47
48