Download PHY2100 Physics Practical II

PHY2100 Physics Practical II
UNIVERSITY OF MALTA
MSIDA, MALTA
2 September 2013
2
CONTENTS
Contents
Experiment 1 - The Specific Charge of an electron by the
Magnetron Method
3
Experiment 2 - The Hall Effect
7
Experiment 3 - The Skin Effect
11
Experiment 4 - Physics with a car headlamp
14
Experiment 5 - Determination of the ratio of the specific
heat capacities of air by the Clément and Desormes’
method
20
Experiment 6 - Forbe’s Bar
23
Experiment 7 - Measuring the linear expansion of solids as
a function of temperature
26
Experiment 8 - The Continuous-Flow Calorimeter
29
Experiment 9 - Verification of Cauchy’s Formula
32
Experiment 10 - The Balmer Series
34
Experiment 11 - The compound pendulum A
37
3
EXPERIMENT 1 - THE SPECIFIC CHARGE OF AN ELECTRON BY THE MAGNETRON
METHOD
Experiment 1 - The Specific Charge of an electron
by the Magnetron Method
Aim
To determine the specific charge of an electron.
Apparatus
Magnetron on base, heater power supply 0-2A with ammeter, anode
power supply 0-30V with voltmeter and milliammeter, solenoid, solenoid
power supply 0-3A with ammeter.
Description
A magnetron is a thermionic valve with cylindical coaxial anode and
cathode. Electrons emitted by the cathode travel radially to the anode
(see Figure 1a), however in the presence of an axial magnetic field the
electron paths become curved (Figure 1b). At a critical value of the
magnetic field, the electron paths just touch the anode (Figure 1c), any
further increase in magnetic field strength will result in the electrons not
reaching the anode (Figure 1d) so the anode current falls to zero. Measurement of this critical field can be used to determine the specific charge.
Figure 1: Electron paths inside the magnetron for different currents.
The conditions for the critical case (Figure 1c) are that the radius of the
electron orbit is half the anode radius of the magnetron, i.e.
re =
ra
.
2
(1)
4
EXPERIMENT 1 - THE SPECIFIC CHARGE OF AN ELECTRON BY THE MAGNETRON
METHOD
But for the the circular motion of an electron with speed v
me v 2
= evBc ,
re
(2)
where e and me have their usual meanings and Bc is the critical magnetic
field.
From conservation of energy
1
eVa = me v 2 .
2
(3)
Combining these equations gives
e
8Va
2Va
= 2 2 = 2 2.
me
Bc re
Bc ra
(4)
Hence, e/me can be found from a graph of Bc2 against Va .
To determine the critical magnetic field Bc from the corresponding solenoid
current Ic , the standard formula is used
B = µ0 In √
l
,
l2 + d2
(5)
where n is the number of turns per metre of the solenoid, d its diameter
and l its length.
Note that µ0 = 4 × 10−7 Hm−1 and the anode diameter of the magnetron
(2ra ) is 6.8mm. The turns-per-unit length of the solenoid are given on
the solenoid and the other dimensions required can be measured using
suitable instruments.
Procedure
The magnetron should be connected as shown below so that
• a current of up to 2A is passed through its cathode filament
• a variable voltage of up to 30V can be maintained between its anode
and cathode, and the current in this circuit measured.
5
EXPERIMENT 1 - THE SPECIFIC CHARGE OF AN ELECTRON BY THE MAGNETRON
METHOD
Figure 2: Circuit set-up for the experiment.
• the solenoid, with its own power supply, can be placed over the
magnetron to provide a variable axial magnetic field.
To emit electrons, the filament should be white-hot. Apply current until the filament glows brightly (∼ 1.6 A) but do not exceed 2A. To
maximise the life of the tube, do not leave it with the filament heated
unnecessarily, i.e. turn down the filament current when not taking readings.
As the anode voltage Va is turned up, an anode current Ia will be read
on the milliammeter which increases as the anode voltage is increased.
In this experiment, the anode voltage is kept fixed while the magnetic
field is varied.
With the solenoid placed over the magnetron, turn up the solenoid current Is . It will be noted that above a certain value the anode current
falls to zero. The aim of this experiment is to determine this critical
magnetic field for different values of the anode voltage.
Note The solenoid quickly heats up with the current passing through
it. This changes its resistance and hence the current flow if the power
supply is used in its usual (voltage-control) mode. Better results can
be obtained by using the power supply in current-control mode. This is
done by turning the current control to zero and the voltage control to
6
EXPERIMENT 1 - THE SPECIFIC CHARGE OF AN ELECTRON BY THE MAGNETRON
METHOD
maximum, then the current control can be turned up to set the desired
current. Caution: Do not pass high currents through the solenoid for
longer than necessary.
To perform the experiment, the anode voltage Va is set to a number of
values between 3V and 20V (chosen with good judgement by the student)
and for each setting, a graph of anode current Ia against solenoid current
Is plotted. In practice, the anode current does not drop suddenly to
zero at the critical current Ic , but falls gradually. Hence the critical
current must be determined graphically. This is done by extrapolating
the two straight-line segments of the curve and determining their point
of intersection, as in Fig.(3).
Figure 3: Graph of anode current to solenoid current. The critical current occurs at
the turning point
Further Reading
Grant and Phillips, Electromagnetism, pp 126-128.
7
EXPERIMENT 2 - THE HALL EFFECT
Experiment 2 - The Hall Effect
Aim
To determine the charge carrier density for a number of Hall effect setups.
Apparatus
Electromagnet and power supply, search coil and galvo, mounted n- and
p-type semiconductor chips, electronic millivoltmeter, milliammeter, 6V
batteries, rheostat.
Note: Do NOT substitute a power supply for the batteries and rheostat
since electronic power supplies may generate several millivolts of noise.
For similar reasons , an analogue millivoltmeter will be give better results
than a digital one in this experiment.
Description
An electric current consists of moving charge carriers (typically negatively charged electrons). In the presence of a perpendicular magnetic
field, these charge carriers will be deflected, creating a potential difference perpendicular to both the magnetic field and the current flow. This
is known as the Hall effect.
Figure 1: Set-up for the Hall effect.
The magnitude of the effect is inversely proportional to the number of
charge carriers and so a semiconductor, which has a carrier density some
104 times less than a metal, is used so that the Hall voltage is can be
measured using relatively unsophisticated equipment.
EXPERIMENT 2 - THE HALL EFFECT
8
To convert the galvanometer readings to the magnetic field, it is necessary to know the sensitivity of the galvo and the size and number of turns
of the search coil. The sensitivity of the galvo is 0.007Wb (7×10−3Wb)
full-scale, the search coil has an area of 1cm2 and 100 turns (given as
100cm2 - turns).
The sensitivity of the combination is thus 7 × 10 − 5 Wb cm−2 or 0.7 Wb
m−2 full-scale. The SI unit of magnetic field, the Tesla, is equivalent to
1 Wb m−2 , so this corresponds to a sensitivity of 0.7T full-scale.
The current flow I in a conductor with n carriers of charge e per unit
volume is given by
I = nAve,
(1)
where A is the cross-sectional area of the conductor and v the drift velocity of the carriers.
The force exerted by a magnetic field on the charge carriers is
FB = evB =
IB
.
nA
(2)
As the charge carriers move under this force, they build up an electric
field which opposes this motion. The magnitude of this force is
FE = Ee =
EVH
,
d
(3)
where d is the thickness of the chip.
At equilibrium, these forces are equal so
IB
EVH
=
,
nA
d
(4)
or
VH =
d
IB.
neA
(5)
Since all other quantities are known (or can be measured), the sign of e
and the carrier density n can be determined.
9
EXPERIMENT 2 - THE HALL EFFECT
Procedure
The first part of the experiment consists of calibrating the electromagnet. The magnet power supply reads magnet current in Ampéres, while
the Hall voltage is a function of the magnetic field. A calibration graph
therefore has to be drawn up to convert meter reading in amperes to
magnetic field in Tesla.
To do this, a search coil is used. It can be shown that the maximum
deflection of a heavily-damped (ballistic) galvanometer is proportional
to the change of flux through the search coil. Thus, to measure the
magnetic field, all that is required is to set the galvo to zero with the
search coil far from the magnet, then place the search coil in the field
and read the maximum deflection. It is probably easier to zero the galvo
with the coil in the field and take the reading after removing the coil:
the deflection will be of course equal (and opposite).
After applying a suitable conversion factor the magnet calibration curve
can be plotted for use in the second part of the experiment.
Note The magnet heats up when current passes through it, for this reason ensure that the cooling water is running before applying power. Do
not forget to turn off the water when you are finished.
Caution The galvo is a sensitive instrument. Ensure that it is unclamped before use, and clamp it again after use. Do not move the
galvo while it is unclamped.
To measure the Hall effect, the semiconductor chip is placed in the magnetic field and connected as shown in Fig.(2).
Note Do not exceed a current of 50mA through the chip.
The chip has an adjustment to compensate for manufacturing errors.
Since the two side contacts may not be exactly opposite each other, a
potential difference may exist between them even without a magnetic
field. The knob should be adjusted so that the millivoltmeter reads zero
EXPERIMENT 2 - THE HALL EFFECT
10
Figure 2: Circuit set-up for the second part of the experiment.
with no applied field, with the maximum current flowing through the
chip.
Take readings to show that:
(a) the Hall voltage is proportional to the applied magnetic field
(b) the Hall voltage is proportional to the current through the chip
(c) the Hall voltage changes sign between the p-type and n-type semiconductor.
Draw diagrams to show that the sign of the Hall voltage indicates that
the charge carriers are negatively charged in the n-type semiconductor
and positively charged in the p-type semiconductor.
Further Reading
Grant and Phillips, Electromagnetism, pp 126-128.
Blakemore, Solid State Physics, pp 166-167.
EXPERIMENT 3 - THE SKIN EFFECT
11
Experiment 3 - The Skin Effect
Aim
To measure the electrical conductivity of copper.
Apparatus
Skin-effect apparatus, signal generator, current probe, oscilloscope.
Description
The solutions of Maxwells Equations in a current-carrying conductor
have an exponential decay with depth inside a conductor. This leads to
the skin effect, whereby a high-frequency alternating current in a wire is
carried essentially only in a thin skin near the outer surface of the wire,
while no current flows through the centre of the wire.
Theory indicates that for sufficiently thick wires (ie. the wire surface
can be considered as an infinite flat plane) the distribution of current
varies exponentially with depth z
I = I0 e−z/δ ,
(1)
where δ is the skin depth, the depth at which the current falls to 1/e
of its value at the surface. Thus, graphs of log I against z should be a
straight line of slope −1/δ.
The dependence of δ on frequency is given by the relation
r
2
,
δ=
µµ0 σω
(2)
where µ is the permittivity of the material (about 1), σ its conductivity
and ω the angular frequency (= 2πf ). A further graph should be drawn
to verify this relation and obtain a value for σ which can be compared
to the standard value for copper obtained from tables.
As a guide, the skin depth for copper is about 1cm at 50Hz, decreasing
with the square-root of the frequency to less than 10-3cm at 50MHz.
12
EXPERIMENT 3 - THE SKIN EFFECT
Figure 1: The compound conductor interior.
Procedure
The apparatus consists of a composite wire made up of several smaller
wires packed together (see diagram). Over a short distance in the centre, the individual wires are separated so that the current in each can be
measured using the current probe. The current probe is a ferrite toroid
that can be clipped round a wire. The magnetic field caused by the
current in the wire induces a proportional voltage in a coil wound on the
toroid, which can be measured using the oscilloscope. The other channel
of the oscilloscope should be connected directly to the signal generator
and used for triggering the timebase, thus ensuring stable display even
with a small noisy signal.
13
EXPERIMENT 3 - THE SKIN EFFECT
The procedure is thus to set the signal generator to a given frequency
and pass as large a current as possible through the apparatus. The apparatus appears essentially as a short-circuit across the signal generator,
which is thus not operating in an ideal situation. The current should
not be turned up beyond the point at which distortion becomes evident.
At a fixed frequency, the current in wires at different distances from
the surface of the bundle can then be measured. Although the current
probe is calibrated (in Volts output per Ampére), this is not needed in
this experiment since it is just a constant of proportionality. Several
wires at the same “depth” should be sampled to obtain an average. The
procedure can then be repeated at different frequencies.
Further Reading
This experiment is based on an article in American Journal of Physics
Vol 44 no. 10 (October 1976), pp 978-980, and was constructed by Dr.
Pauline Galea while a B.Sc. student as her final-year project.
A derivation for the formula for the skin depth can be found in Grant
and Phillips, Electromagnetism, pp 379-383.
EXPERIMENT 4 - PHYSICS WITH A CAR HEADLAMP
14
Experiment 4 - Physics with a car headlamp
Aim
To determine the efficiency of a tungsten filament light bulb.
Apparatus
12V Digital power supply unit (p.s.u.), 60/50 W car headlamp, thermometer.
Background
In this experiment, you will try to answer some real-life questions on the
tungsten filament of a car headlamp, such as
• How efficient is the halogen gas-filled lamp with a tungsten filament?
• What temperature does the filament reach?
• In what part of the electromagnetic spectrum is most of the radiation emitted from the hot lamp?
In order to answer the above questions, one needs to do a lot of data
analysis, and hence in this experiment, you will realise how spreadsheet
programs, are indispensable for large amounts of data manipulation.
Description
When a lamp has reached a steady state, the electrical power dissipated
in the filament must be equal to the total rate of heat lost from the
filament, due to conduction, convention and radiation. However, when
the lamp is glowing, we can safely assume that most of the energy is
lost by radiation and so the electrical power dissipated in the filament,
Pel = V I, is effectively equal to the power lost by radiation, Prad .
In the first part of the experiment you will check whether the empirical
formula
m
R
T = Ts
,
(1)
Rs
EXPERIMENT 4 - PHYSICS WITH A CAR HEADLAMP
15
is valid, where R is the resistance at the temperature T and Rs is the
resistance at the room temperature Ts . Thus, using Eq.(1), one can
estimate the temperature T of a halogen lamp at a given resistance
R. Incidentally, m is a constant whose value can be found from the
experimental results as explained below.
To find m
If we assume that the hot tungsten filament behaves as a greybody the
net power radiated, to a good approximation, is described by the StefanBoltzmann law
Prad = Aσ T 4 − Ts4 ,
(2)
where the filament is at a temperature T and the surrounding room
temperature is Ts is the emissivity of the filaments surface, A is the area
of the surface from which the radiation is emitted and σ is the StefanBoltzmann constant. For a blackbody surface, = 1; for other surfaces,
the emissivity is a complicated function of temperature, environment
and fabrication (Incropera and De Witt,1990). However, to simplify the
analysis the following assumptions are made about the filament
• Its emissivity is constant and independent of the temperature,
• It has a uniform surface temperature,
• All the radiation is emitted from its surface.
For the filament, the constants A and are unknown. However, these
two unknown quantities can be eliminated by taking a ratio. If P1 is the
power when the filament just starts to glow brightly at a temperature
T1 (T14 < Ts4 ) and Pi is the power at a higher temperature Ti then,
4
Pi
Ti
=
.
(3)
P1
T1
Combining Eqs(1, 3), it is possible to relate the power radiated to the
resistance of the filament
Pi
Ri
log
= m 4 log
.
(4)
P1
R1
If the predictions given by Eqs.(1,2) are valid, a plot of log(Pi /P1 ) against
4 log(Ri /R1 ) will yield a straight line with slope m through the origin.
EXPERIMENT 4 - PHYSICS WITH A CAR HEADLAMP
16
In this way the temperature the filament reaches can be calculated. But
what about the efficiency of the car headlamp? We will try to answer
these questions in the second part of the experiment.
Tungsten Efficiency
Assuming that the tungsten filament radiates as a greybody, then the
spectrum of the emitted radiation will be given by the Planck distribution function (Eisberg and Resnick 1985)
p (λ) dλ =
N
dλ,
λ5 ehc/λkT − 1
(5)
where p (λ) dλ is the radiant power emitted in the wavelength interval
λ and λ + dλ with p(λ) being the power emitted per unit wavelength
in this interval, N is a normalising factor, λ is the wavelength of the
emitted radiation, h is the Planck constant, c is the speed of light and
k is the Boltzmann constant. The integral over all wavelengths of the
function p(λ) given by Eq.(5) equals the total power radiated by the hot
filament at a temperature T , i.e.
Z
Prad = p (λ) dλ.
(6)
With a spreadsheet this integral can be evaluated numerically and the
value of N adjusted so that the integral is equal to the input power, Pel .
The luminous efficiency of the lamp is defined as the ratio of the power
radiated in the visible part of the spectrum to the input power. Thus,
the efficiency of the lamp can be found by comparing the ratio of areas
under the Planck distribution curve.
Procedure
First note the room temperature Ts and then set up the following simple
circuit. Using a digital power supply unit one can vary the voltage across
the lamp and then read the corresponding current.
• Start increasing slowly the voltage across the lamp until the filament lamp just starts to glow brightly. Note the voltage V1 (approximately about 1V) and the corresponding current I1 .
17
EXPERIMENT 4 - PHYSICS WITH A CAR HEADLAMP
Figure 1: Circuit set-up for the experiment.
• Afterwards, take about 60 current readings for voltages between
0.2V-12V, i.e. in 0.2V steps. As the voltage is increased it is necessary to wait before the readings are recorded so that an equilibrium
is established.
• Finally, take 3 more readings in the range 13V-15V, (i.e. in 1V
steps).
The steps for the data analysis of the experiment are as follows
1. Input the data in some Worksheet program and hence plot the I-V
characteristic of the lamp.
2. For each pair of values, calculate the resistance R (= V /I) of the
filament and the dissipated power P (= V I). Plot an R-I graph.
3. The room temperature resistance Rs can now be found by extrapolating the R-I graph to find the resistance at zero current. Note
you might notice some deviation from linearity for small values of
the resistance. In order to handle this situation the points that
deviate from linearity will have to be ignored. These points should
be clearly marked on the graph (Hint: make two data series and
use a legend). State the reason(s) that caused the deviation from
linearity that has been observed in the discussion.
4. Thenext
calculate
the constant m. Verify the relationship
step is to log PP1 = 4m log RR1 and hence calculate the value of m using
linear regression analysis tools.
EXPERIMENT 4 - PHYSICS WITH A CAR HEADLAMP
18
5. Now, the temperature T (in Kelvin) of the filament is related to its
resistance R by
m
R
T = Ts
.
(7)
Rs
Since Ts , Rs and m are now known, plot a graph of R vs. T and
comment.
6. In the final part of this experiment you will try to calculate the
luminous efficiency of the lamp. Recall, that if we assume that the
Tungsten filament lamp radiates as a grey body, the spectrum of
the emitted radiation is given by Eq.(5), i.e.
p (λ) dλ =
N
λ5 ehc/λkT − 1
dλ.
(8)
(a) Set up a column of about 500 wavelength values ranging from
200nm to 9200nm, i.e. ( dλ ' ∆λ = 18nm).
(b) Let T be the maximum temperature of the filament (i.e. at
15V). For each value of λ, calculate the right hand side of
Eq.(5). The value of the normalising constant N should be
adjusted to make the area under the graph equal to the power
dissipated at that temperature. (Hint: Find the area of the
rectangle subtended by each dλ and sum).
(c) Plot a graph of Power / wavelength interval (W/nm) vs. wavelength (nm).
(d) Sum the values of p(λ)dλ for λ values in the range 400nm 700nm (visible range).
Sum all the values of p(λ)dλ (i.e. from 200 - 9200nm)
Express these two sums as a percentage ratio. This percentage
ratio is the luminous efficiency of the car headlamp.
Comment on your result.
Useful Constants
Plancks constant (h)
= 6.62608 × 10−34 Js,
Boltzmanns constant (k) = 1.38066 × 10−23 JK1 ,
Speed of light (c)
= 2.99792 × 108 ms−1 .
19
EXPERIMENT 4 - PHYSICS WITH A CAR HEADLAMP
Further Reading
F. S. Levin, An Introduction to Quantum Theory. Cambridge University
Press, 2002.
20
EXPERIMENT 5 - DETERMINATION OF THE RATIO OF THE SPECIFIC HEAT
CAPACITIES OF AIR BY THE CLÉMENT AND DESORMES’ METHOD
Experiment 5 - Determination of the ratio of the
specific heat capacities of air by the Clément and
Desormes’ method
Aim
To find the adiabatic constant of air.
Apparatus
The apparatus is assembled.
Description
Let the v1 be the specific volume of air at the ambient temperature T0
and at a high pressure P1 . If the gas is expanded adiabatically to the
atmospheric pressure P0 its new specific volume, v0 , would be given by
P1 v1γ = P0 v0γ ,
(1)
where γ is the ratio of the specific heat capacity of the gas. Such an
expansion would have decreased the temperature of the gas. However
on letting the gas to get back to the ambient temperature T0 at constant
volume a new pressure P2 would result. Considering the isothermal
process that occurs on going from the initial stage to the final stage (i.e.
ignoring the adiabatic expansion), the following equation is obtained
P1 v1 = P2 v0 .
Therefore using Eq.(1)
P0
=
P1
V1
V0
(2)
γ
,
(3)
and using Eq.(2)
V1
P2
= .
V0
P1
Eliminating v0 and v1 from Eqs(3,4) gives
γ
P0
P2
=
.
P1
P1
(4)
(5)
21
EXPERIMENT 5 - DETERMINATION OF THE RATIO OF THE SPECIFIC HEAT
CAPACITIES OF AIR BY THE CLÉMENT AND DESORMES’ METHOD
Taking logarithms on both sides
P0
P2
= γ log
.
log
P1
P1
(6)
Making γ subject of the formula
γ=
log (P0 /P1 )
.
log (P2 /P1 )
(7)
In the case were the pressure difference is very small then an alternative
approach can be taken. If h1 is the initial difference in the levels of the
manometer and h2 the final difference in the levels, we have
P1 = P0 + ρgh1 ,
P2 = P0 + ρgh2 .
Thus
and
P1 − ρgh1
ρgh1
P0
=
=1−
,
P1
P1
P1
(8)
P2
P1 − ρg (h1 − h2 )
ρg (h1 − h2 )
=
=1−
,
P1
P1
P1
(9)
which implies that Eq.(5)
ρgh1
1−
=
P1
γ
ρg (h1 − h2 )
1−
.
P1
(10)
If it is assumed that ρg (h1 − h2 ) << P1 then the expansion
1−
ρg (h1 − h2 )
ρgh1
≈1−γ
,
P1
P1
(11)
can be taken.
Hence
1
h2 = 1 −
h1 .
γ
(12)
Repeat the experiment for six different values of h1 and each time finding the corresponding value of h2 . (Note no repeated readings will be
requested in this case) Hence γ can be found.
22
EXPERIMENT 5 - DETERMINATION OF THE RATIO OF THE SPECIFIC HEAT
CAPACITIES OF AIR BY THE CLÉMENT AND DESORMES’ METHOD
Procedure
Close the stopcock which is directly open to the atmosphere. Pump
some air inside the carboy until the pressure on the manometer is 15 cm
of water above atmospheric. Allow fifteen minutes to elapse so that the
temperature of the air inside the bulb reaches the outside temperature.
Open the same stopcock suddenly (about 1/2 s once the pressure begins
to change, i.e. the manometer level begins to move) until the pressure
inside becomes atmospheric and close the stopcock again. Obviously
the levels of the water in the manometer will be the same at first, but a
rise in temperature due to the passage of heat from the outside into the
vessel (the air inside having been cooled due to the sudden expansion)
will result in an increase in the pressure inside the carboy. This will
increase the manometer levels until they become steady. Record the
final difference in the levels together with the original pressure inside
the carboy.
Further Reading
Francis Weston Sears and Gerhard L. Salinger, Thermodynamics, Kinetic Theory, and Statistical Thermodynamics, Third Edition.
23
EXPERIMENT 6 - FORBE’S BAR
Experiment 6 - Forbe’s Bar
Aim
To determine the thermal conductivity of copper.
Apparatus
Copper bars (Forbes bar, and a calibration bar), boiler, two hotplates,
thermocouple and millivoltmeter, vacuum flask with ice, beaker, mercury
thermometer.
Description
A copper rod has one end held at 100◦ C by means of a steam jacket. At
steady state, a temperature gradient is established along the rod that
can be measured using thermocouple. Analysis of the results enables the
thermal conductivity k of the rod to be determined.
The temperature gradient along the bar in steady state satisfies the
continuity equation
d2 θ
(1)
kA 2 = P H,
dx
where
k is the thermal conductivity of the material,
A the cross-sectional area of the bar,
P the perimeter of the bar,
H is the rate of heat loss per unit length of the bar.
Integrating the above equation
x 2
Z x2
Z
dθ
mc x2 dθ
kA
=P
H dx = P
dx,
dx x1
S x1 dt
x1
(2)
where x1 and x2 are tow distances along the bar and m, c and S are
respectively the mass, specific heat and surface area of the calibration
bar which can be assumed to have the same heat loss characteristics as
the large bar (note that A and P refer to the large bar).
24
EXPERIMENT 6 - FORBE’S BAR
The Left-Hand-Side of the equation can be evaluated by drawing tangents to the curve of θ against x at two positions x1 and x2 , and finding
the difference in their slopes.
The Right-Hand-Side (RHS) of the equation can be evaluated by drawing tangents to the curve of θ against t and plotting a graph of (dθ/dt)
against x (the corresponding values of x for each t can be obtained from
the first graph). The integral in the RHS of the equation is then given
by the area under the curve of this third graph, between the positions
x1 and x2 .
In the above equation the only unknown is the thermal conductivity k
of the bar which can thus be found.
Procedure
Since it takes a long time (at least 2 hours) for the bar to reach thermal
equilibrium, the first step is to ensure that the steam generator is full
of water and to turn it on. Aim to keep a steady flow of steam through
the apparatus throughout the experiment, but do not let the boiler run
dry. Top up (carefully) when required.
While the bar is warming up, the thermocouple should be calibrated.
First check the thermocouple wires and their connection to the millivoltmeter. Check also that the cold junction is held securely in place in
an ice-water mixture, the use of a vacuum flask means that the ice will
last the duration of the experiment. Check the polarity by warming the
hot junction with your fingertips - if the meter reading decreases, change
the junctions over or connect the meter in the opposite polarity.
To calibrate the thermocouple, place the hot junction in a beaker of water
on the second hotplate, together with the mercury-in-glass thermometer. Note the thermocouple output at various temperatures between
room temperature and 100◦ C.
Before commencing the experiment, it is important to ensure that the
bar has reached a steady state. This means that the temperature at
EXPERIMENT 6 - FORBE’S BAR
25
any point on the bar does not change over an appreciable period of time
(several minutes). Once steady state is reached, the temperature θ can
be measured as a function of position x along the bar, and a graph plotted.
The final step is to obtain a measure of the rate of heat loss from the
bar. This is done by heating the calibration bar to above the highest
temperature recorded, and letting it cool in a similar environment while
measuring its temperature φ as a function of time t, and a graph plotted.
The length, diameter and mass of the calibration bar should be measured as well as the diameter of the Forbes bar, and the specific heat of
copper obtained from a data book.
The steps for the data analysis of the experiment are as follows
1. Sketch the calibration graph of change in temperature against voltage. Should be linear?
2. Sketch a graph of change in temperature against x. How should
this fall off as?
3. Sketch a graph of change in temperature against time for the calibration bar.
4. Finally use the x positions in the second graph to correlate dθ/dt
with the x positions. For each position find the respective change
in temperature through the second graph then using the gradient
of the third graph find dθ/dt. Lastly sketch the graph dθ/dt.
5. Determine the thermal conductivity k of copper using Eq.(2)
Further Reading
Francis Weston Sears and Gerhard L. Salinger, Thermodynamics, Kinetic Theory, and Statistical Thermodynamics, Third Edition.
26
EXPERIMENT 7 - MEASURING THE LINEAR EXPANSION OF SOLIDS AS A FUNCTION
OF TEMPERATURE
Experiment 7 - Measuring the linear expansion of
solids as a function of temperature
Aim
Determining the linear expansion coefficients of brass, steel and glass.
Apparatus
Longitudinal expansion apparatus, Dial gauge, Circulation thermostat
with pump and thermometer, water tubing, glass, steel and brass tubes,
small beaker, Mercury thermometer, Measuring tape.
Theory
In response to a change in temperature a body will change its dimensions.
For a protruding dimension of length l consider the linear dependence
of the rate of change of length with respect to temperature per unit
(reference) length
1 ds
= α,
(1)
l0 dθ
where l0 is the original length (at some reference temperature). Solving
this gives the length at any temperature θ1 as
l1 = l0 (1 + α∆θ) ,
(2)
where ∆θ = θ1 − θ0 .
Now the constant α does depend on the material under investigation,
not the original length of the material under investigation. It should
also be noted that this is an approximation, if the readability interval
of the measuring device were rendered smaller then higher order terms
would become significant as well.
In this experiment a circulation thermostat is used to heat the water
which flows through the various tube samples. A dial gauge with 0.01
mm scale graduations is used to measure the change of length ∆l a
function of temperature θ.
27
EXPERIMENT 7 - MEASURING THE LINEAR EXPANSION OF SOLIDS AS A FUNCTION
OF TEMPERATURE
Figure 1: Schematic representation of the experimental setup to measure the linear
thermal expansion of tubes with the expansion apparatus as a function of temperature.
Using the Circulation Thermostat Unit
The unit contains a pump, thermometer and a heating component. The
pump should NOT be switched on unless the water level is above the
pump inlet (that is more than half way full). If any warming signals appear switch of the device and seek out one of the laboratory attendants.
Once the in- and out-lets have been connected to the tube under investigation and the forward tank filled with water the unit can be switched
on. This is achieved by turning on the power switch located on the back
of the unit as well as the power button on the front side of the unit. The
temperature can be changed by unit the arrows and selecting with temperature on the control screen with the center (control) button. Using
the arrows the value can be changed and then confirmed by pressing the
center button again.
In order to take readings wait for the temperature to settle as much as
it can as well as eliminating as many environmental sources as possible.
Procedure
1. Carefully secure the brass tube onto the longitudinal expansion
apparatus. This means screwing in the tube such that the dial
reading moves slightly.
2. Measure the length of the brass tube with the measuring tape and
28
EXPERIMENT 7 - MEASURING THE LINEAR EXPANSION OF SOLIDS AS A FUNCTION
OF TEMPERATURE
the room temperature with the thermometer.
3. Take note of the dial reading, this will be the reference value used
in the change of length calculation.
4. Use the Circulation thermostat to pump heated water through the
tube under investigation. Length and temperature should be noted
every 5◦ C in the range 35◦ C − 95◦ C. The real temperature should
be noted as opposed to the target temperature.
5. Take repeated 3 readings (first raising the temperature, then decreasing it, and finally increasing it again). In order to decrease the
temperature it may be needed that water is exchanged with supply
water.
6. Plot a graph of change in length against change in temperature and
determine the α coefficient. Compare this to community values.
7. The process should be repeated for the steel and glass tubes.
Further Reading
Francis Weston Sears and Gerhard L. Salinger, Thermodynamics, Kinetic Theory, and Statistical Thermodynamics, Third Edition.
29
EXPERIMENT 8 - THE CONTINUOUS-FLOW CALORIMETER
Experiment 8 - The Continuous-Flow Calorimeter
Aim
To determine the specific heat capacity of water.
Apparatus
Callender and Barnes calorimeter, constant-head device, two 0.1C mercury thermometers, 4A power supply with ammeter and voltmeter, stopwatch, measuring cylinder.
Description
Calorimetry as a means of measuring the specific heat of a substance is
complicated by the effects of heat losses. In a continuous-flow calorimeter, two measurements can be taken at identical temperatures, thus enabling the effects of heat losses to be cancelled out.
In its simplest form, the calorimeter consists of a tube through which
water flows at a known rate. Heat is added to the water by an electric
heater, also at a known rate, and the temperature rise measured. The
rate of flow is then altered and the electrical power input varied to obtain
the same inlet and outlet temperatures as before. From the two sets of
results, the specific heat of water can be determined to an accuracy set
essentially by the precision of the thermometers used.
If m is the mass of water of specific heat capacity c that flows through
the calorimeter in unit time, then the temperature change expected is
given by
IV = mc∆θ,
(1)
where IV is the electrical power input, ∆θ the temperature change across
the apparatus and H the rate of heat loss.
If the experiment is repeated with the same mean temperature and hence
30
EXPERIMENT 8 - THE CONTINUOUS-FLOW CALORIMETER
the same rate of heat loss, then for the two experiments
I1 V1 = m1 c∆θ1 ,
I2 V2 = m2 c∆θ2 ,
(2)
(3)
from which H can be eliminated and c determined.
Procedure
This experiment has two variables that can be adjusted, the flow rate
and the electrical power. Before starting the experiment, make an estimate of the required flow rate to give an acceptable rise in temperature,
given that the current in the heater should not exceed 4A. If the flow
rate is too high, the temperature difference will be low compared to the
precision of the thermometers. Remember that you also need to perform
the experiment at a second flow rate, which can only be lower than the
first since the electrical power input cannot be increased further.
Set up the apparatus and measure the flow rate by measuring the amount
of water flowing through the apparatus in a known time. The flow rate
can be varied by altering the height of the constant-head apparatus, or
by applying a clip to one of the hoses.
After switching on the heater, ensure that the apparatus has reached
steady state before recording the current, voltage and temperature readings. This can be checked by seeing that the temperature readings do
not vary over a period of several minutes.
Reduce the flow rate and repeat the experiment, adjusting the electrical
power input to get the inlet and outlet temperatures as close as possible
to those in the first experiment. If the inlet temperature changes, aim
to get the average of the two temperatures as close as possible to the
average in the first experiment.
Using a linear combination of Eqs.(2, 3) the specific heat capacity can
be determined. (This experiment has no associated graph).
31
EXPERIMENT 8 - THE CONTINUOUS-FLOW CALORIMETER
Further Reading
Francis Weston Sears and Gerhard L. Salinger, Thermodynamics, Kinetic Theory, and Statistical Thermodynamics, Third Edition.
32
EXPERIMENT 9 - VERIFICATION OF CAUCHY’S FORMULA
Experiment 9 - Verification of Cauchy’s Formula
Aim
To determine the material under investigation through a comparison
table and the determination of the Cauchy constants.
Apparatus
Spectrometer, prism, variety of spectral lamps (sodium, mercury and
cadmium suggested).
Description
In general the variation of the refractive index µ of a material as a
function of the wavelength λ can be expressed as a polynomial expansion
µ=A+
C
B
+
+ ... =
λ2 λ4
X
n∈{0,1...}
Ci
,
λ2n
(1)
provided that the wavelength is well away from an absorption band (i.e.
the material is transparent at that wavelength). Often the first two
terms are sufficient to obtain good agreement with experimental results,
leading to Cauchys Formula
µ=A+
B
.
λ2
(2)
The aim of this experiment is to verify this relationship and determine
the value of the constants A and B.
If δm is the angle of minimum deviation (change in angle between ingoing
and outgoing rays) and α the apex angle of the prism (angle subtended
opposite shaded side), then the refractive index µ is given by
m
sin α+δ
2
µ=
.
sin α2
(3)
33
EXPERIMENT 9 - VERIFICATION OF CAUCHY’S FORMULA
Spectral lines
Sodium
(yellow)
589.6 nm (close double)
Mercury
(yellow)
578.0 nm (close double)
(green)
546.1 nm
(blue)
491.6 nm (weak)
(violet)
435.8 nm
Cadmium
(red)
643.8 nm
(blue-green)
508.6 nm
(blue)
480.0nm
Procedure
Mount the glass prism on the spectrometer table and set up the spectrometer (refer to your first year lab notes if necessary). Then, using
the sodium source, determine the apex angle of the prism. Still using
the sodium source, determine the angle of minimum deviation at that
wavelength.
Repeat the measurement of the angle of minimum deviation with various
spectral lines from other spectral lamps. A selected list of wavelengths
is given below, however other lamps may be used although their wavelengths will have to be looked up in a data book.
From the angles of minimum deviation, calculate the refractive index at
each wavelength. Then draw a suitable graph to verify Cauchy’s formula
and determine the values of the coefficients A and B. Hence deduce the
material under investigation.
Further Reading
Jenkins and White, Fundamentals of Optics, Fourth Edition. pp 3032 (angle of minimum deviation) pp 438-456 (sources of light and their
spectra) pp 474-482 (dispersion)
34
EXPERIMENT 10 - THE BALMER SERIES
Experiment 10 - The Balmer Series
Aim
To compare the experimentally obtained Rydberg constant to the one
derived from quantum theory.
Apparatus
Spectrometer, diffraction grating, hydrogen and deuterium spectral lamps.
Description
In 1855, Balmer obtained an empirical relation for the wavelengths of
the spectral lines of hydrogen in the visible region
1
1
1
= RH
−
n = 3, 4, 5,
(1)
λ
22 n2
where RH is known as the Rydberg constant for hydrogen and has the
standard value of 1.1 × 107 m−1 .
The major success of the Bohr theory of the hydrogen atom (1913) was
in explaining this seemingly arbitrary formula.
In this experiment, the wavelengths of the visible spectral lines of the
hydrogen spectrum are determined using a grating spectrometer, so as
to verify the above formula and obtain an experimental value for RH .
Diffraction grating formula
nλ = d sin θ.
(2)
The Bohr theory gives the radius of the nth atomic orbital of an atom of
proton number Z as
4π0 n2 ~2
,
(3)
rn = 2
e Zme
where me is the mass of the electron and the other symbols have their
usual meaning (~ = h/2π).
EXPERIMENT 10 - THE BALMER SERIES
35
The energy associated with this orbit is
Ze2
E=−
,
8π0 r
whence
En = −
e2 Z
2
me
(4π0 )2 2n2 ~2
(4)
,
(5)
so that the energy difference between two levels n and m (n > m), and
thus the energy of a photon emitted by a transition between these two
levels, is given by
e4 Z 2 me
1
1
1
hc
1
=
− 2 = Z 2 RE
−
, (6)
En − Em =
2
2
2
λ
n
m2 n2
2~ (4π0 ) m
so that
e4 me
RE =
.
2~2 (4π0 )2
(7)
What does this give for the Rydberg constant? How is this an explanation of the phenomenon?
Transitions to the innermost orbit (m = 1) have energies in the ultraviolet region and are named the Lyman series after their discoverer.
The Balmer series correspond to transitions to the second orbital, with
m = 2.
Note also that the Rydberg constant RH is expressed in terms of only
fundamental constants in the quantum theory.
Procedure
Mount the diffraction grating on the spectrometer table and set up the
spectrometer (refer to your first year lab notes if necessary). The lines
emitted by the hydrogen lamp are very dim, so in performing the experiment ensure that:
(a) the room is as dark as possible
(b) the spectral lamp is well shielded to avoid stray light
(c) the spectral lamp has reached its operating temperature and hence
maximum brightness - this can take up to 15 minutes
36
EXPERIMENT 10 - THE BALMER SERIES
(d) your eyes are well accustomed to the darkness. This can also take
several minutes.
You should be able to see at least three spectral lines in the first order
(red, blue-green, deep blue and possible violet), and perhaps one or two
in the second order. Take as many measurements as you can since the
accuracy of your calculations can be improved by taking the average of
several readings.
Having calculated the wavelengths of each line (the red corresponds to
n = 3), plot a suitable graph to verify Balmers formula and obtain a
value for RH using both the gradient and intercept as the two experimentally obtained Rydberg constants.
If time permits, repeat the experiment with the Deuterium (2 D) source.
What difference (if any) would you expect to see in your results?
Further Reading
Semat and Albright, Introduction to Nuclear Physics, Fourth Edition.
pp 219-232
Jenkins and White, Fundamentals of Optics, Fourth Edition. pp 611-631
37
EXPERIMENT 11 - THE COMPOUND PENDULUM A
Experiment 11 - The compound pendulum A
Aim
To find a number of values of the radius of gyration for a compound
pendulum.
Apparatus
Compound pendulum, stop watch, pivot.
Theory
Consider the uniform rigid rod shown in Fig.(1), where the centre of mass
is at G and the point of suspension is at O such that the rod can swing
freely in the vertical plane about a horizontal axis through O. When the
rod is displaced by an angle (φ) from the vertical, the restoring torque
(τ ) is given by
τ = −mgh sin φ.
(1)
Figure 1: Experimental set-up.
If the rod is now released it will oscillate. The angular acceleration
EXPERIMENT 11 - THE COMPOUND PENDULUM A
38
(d2 φ/dt2 ) is given by
d2 φ
τ = −I 2 ,
(2)
dt
where I is the moment of inertia of the rod about O. Now if I is the
moment of inertia of the rod about an axis through G, such that I0 =
mk 2 , where k is the radius of gyration of the rod about G, then by the
parallel axes theorem
I = I0 + mh2 .
(3)
Thus
I = m k 2 + h2 .
(4)
Now, from Eqs.(2, 3), the governing equation of motion is
d2 φ
+ gh sin φ = 0,
dt2
d2 φ gh sin φ
+
= 0.
dt2
I
I
(5)
(6)
For small angles of φ, Eq.(6) may be approximated to
d2 φ ghφ
+
= 0.
dt2
I
This describes S.H.M. with period T given by
s
I
T = 2π
,
mgh
and on substituting for I from Eq.(6)
s
k 2 + h2
T = 2π
.
gh
(7)
(8)
(9)
Now, for a simple pendulum, the period for small oscillations is given by
s
l
T0 = 2π
.
(10)
g
Then, a simple pendulum with period T0 = T for a compound pendulum
would have a length l0 which may be obtained by comparing Eq.(9, 10)
l0 =
k 2 + h2
,
h
(11)
EXPERIMENT 11 - THE COMPOUND PENDULUM A
39
where l0 is the simple pendulum equivalent length (i.e. a simple pendulum length that gives a period equal to that of a compound pendulum
at a given h). We can write Eq.(11) as
h2 − l0 h + k 2 = 0.
(12)
Thus, there will be two values of h that give the same period with an
associated equivalent length lo. Let these be h1 and h2 such that these
are the roots of Eq.(12). Then
q
1
1 2
h1 = l0 +
l0 − 4k 2 ,
(13)
2
2
and similarly
1
1
h2 = l0 −
2
2
q
l02 − 4k 2 ,
(14)
which implies that
h1 h2 = k 2 and h1 + h2 = l0 .
As shown in Fig.(2), if T is plotted against h two branches are obtained,
one corresponding to O placed between A and G and the other corresponding to O placed between B and G.
Figure 2: Expected graph of period against distance from pivot.
From Eq.(9), with h = 0 (i.e. O on G), T is infinite. From Eq.(13,14)
the least value of l0 for real roots is 2k and this corresponds to minimum
T (Fig.(2)) It is seen from Fig.(2) that any horizontal line drawn above
the minima intersects the curves at four points corresponding to h1 , h2 ,
h2 and h1 respectively for a given value of T .
EXPERIMENT 11 - THE COMPOUND PENDULUM A
40
Procedure
1. Find G by balancing the rule on a knife edge
2. Determine T as a function of h by moving the pivot to as many
different points along the rod as possible.
3. Plot a graph of T against h as shown in Fig.(2). Draw a set of
lines parallel to the h-axis across your graph and determine a set of
values of l0 .
4. From your graph determine two values for the radius of gyration
and hence obtain a mean value.
5. Compute another value for the radius of gyration using the following
theoretical relation
L
k=√ ,
(15)
12
where L is the length of the rod.
6. Plot another graph of T 2 h against h2 and obtain two other values
for k and then find an average. Is the rod uniform?
7. Plot a final graph of T 2 h against h2 and determine a further value
for k. Compare all the values that you have found for the radius of
gyration.
Further Reading
Francis Weston Sears and Gerhard L. Salinger, Thermodynamics, Kinetic Theory, and Statistical Thermodynamics, Third Edition.