Download Collection of Problems about RCL "Millikan´s Experiment"

Collection of Problems
about
RCL “Millikan’s Experiment”
S. Gröber
University of Technology Kaiserslautern (Germany)
January 2007
Table of Content
0.
Suggestions for teaching applications and data of experiment
2
I.
Problems about theory
4
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Stokes frictional force
Discrete values
History of Millikan’s experiment
Calculation of values in Millikan’s experiment
Model experiment to the Millikan experiment
Different types of Millikan’s experiment
Selection of oil droplets
Acceleration phase of oil droplets
Cunningham correction
R. A. Millikan
4
4
5
5
5
5
6
6
7
7
II.
Problems about experimental setup
8
1.
2.
3.
4.
5.
Experimental setup of RCL “Millikan’s Experiment”
Observation of oil droplets
Generation of oil droplets
Control of the electric field in the capacitor
Millikan’s original experimental setup
8
8
9
9
10
III. Problems about measurements and data analysis
11
1.
2.
11
11
Performing Millikan’s experiment
Evaluation of Millikan’s experiment
IV. Solutions to problems I
13 - 21
V. Solutions to problems II
22 - 25
VI. Solution to problems III
26
VII. References
27
1
0. Suggestions for teaching applications and data of experiment
1.
Suggestions for teaching applications
To consolidate the usage of this RCL we offer a wide and enduring collection of
problems: either for differently educated, interested students or thematically problems to all components of the experiment, such as theory (I.), experimental setup
(II.), performance and evaluation (III.).
From methodological point of view we offer problems of different categories: pure
problem solving, planning and performing qualitative pre-experiments to prepare
the main one, internet research, students presentations, learning stations within a
learning circle etc.
The following table is containing topics of all problems, so the teacher is getting a
quick survey to the content of each problem. Additionally, we offer suggestions how
to implement each problem into a teaching environment.
No.
Topic
Content
Teaching application
•
I.1
Stokes frictional force
I.2
Discrete values
I.3
History of
Millikan’s experiment
I.4
I.5
I.6
Calculation of values in
Millikan’s experiment
Model experiment to the
Millikan experiment
Different types of
Millikan’s experiment
•
Uniform motion under the influence of
Stokes frictional force
•
Difference between discrete and continuous values
•
Problem of the existence and proof of
the elementary charge
When to teach Millikan’s experiment
•
•
Presentation by students or teacher
Stations in a learning circle
Mathematical-physical relations between magnitudes
Determination of microscopically small
values
•
•
Exercises to deepen theory
Stations in a learning circle
•
Inspection of the qualitative understanding
of Millikan’s experiment
•
Differentiation of students groups according
to their mathematical competence
Competition between students groups in calculating
•
•
•
•
•
•
•
•
I.7
Selection of oil droplets
•
•
•
I.8
Acceleration phase of
oil droplets
I.9
Cunningham correction
I.10
R. A. Millikan
II.1
Experimental set up of
RCL Millikan’s experiment
Preparation of a more independent experimenting with Millikan’s experiment
Introduction of Stokes frictional force
Repetition of motion with friction
Develop and check hypothesis
•
•
•
•
•
•
•
•
Form analogies between the two classes of experiments
Limitations of analogy
Handling of a system of equations in
the physical context
Mathematical transformations
Dependence of results in measurements on selected oil droplets
Make a difference between statistical
and analytical relations
Evaluation of Millikan’s data with respect to new insight
•
•
•
Introduction of the concept and notion of
quantization
Preparation of a more independent experimenting with Millikan’s experiment
Mini research for advanced students
Evaluation of possible acceleration
under the action of varying forces
•
Exact calculation with differential equation
Problems for the application of learned,
known content in a new context
Average free path length and viscosity
of gases
Validity of Cunningham correction
Extreme value of functions
•
Independent acquisition of new learning
context with suited learning materials for students
•
Millikan as a researcher, as a teacher
and as private person
•
•
Presentation by student or teacher
Station in a learning circle
•
Assignment and functions of experimental components
•
As a problem when experimenting with this
RCL for the first time
2
•
•
II.2
Observation of oil drop- •
lets
•
•
•
II.3
Generation of oil droplets
II.4
Generation and control •
of the electric field
•
•
•
•
II.5
Millikan’s original
experimental set up
•
•
III.1
Performing Millikan’s
experiment
•
•
•
III.2
2.
Evaluation of Millikan’s
experiment
•
Difference between dark field and
bright field microscopy
Difference between Rayleigh and Mie
scattering
Measurement of distances by ocular
and object micrometer
Path of optical rays in microscope and
teleobjective
Evaluation of transfer rate of digital video data
•
•
Static and dynamic pressure in liquids,
hydrodynamic paradoxon
•
Bernoulli equation
Evaporation rate of liquids
Calculation of electric field strength
Charging and discharging a capacitor
•
Work with overlapping topics
Internet search with web quest in student
groups
Presentation of teacher
Problems for the application of learnt, known
content in a new context
Difference between experimental set
up as a demonstration experiment and
the original version
•
Wiring for the charge and discharge of
a capacitor
Work with overlapping topics
Setup and test of a spreadsheet for
measuring data
Description of performing the experiment
Explanation for raising and falling oil
droplets
Homework for the preparation of measurements
Examination of the comprehension of theory
Presentation of measured data in a
histogram or point diagram
Determination of error for the charges
according to error propagation rules
•
•
•
Homework for the preparation of evaluation
of measurements
Data of experiment
To solve problems assignments of variables and technical data about Millikan’s experiment can be found in the following tables.
Variables
Mass of oil droplet
Radius of oil droplet
Stokes frictional force
Gravitational force
Electric force
Buoyancy force
Voltage capacitor
Raising voltage
Falling voltage
Floating voltage
Electric field strength
Raising time
Falling time
Raising velocity
Falling velocity
Viscosity of oil
Charge of oil droplet
Corrected viscosity of air
moil
r
Fs
Fg
Fe
Fb
U
Urise
Ufall
Ufloat
E
trise
tfall
vrise
vfall
ηoil
Q
ηcorr,air
Constant values
3
Density of oil droplet
ρoil = 1.03 g/cm3
Distance of capacitor plates
d = 6 mm
Density of air
ρair = 1.3 kg/m3
Viscosity of air
ηair = 1.81·10-5 Ns/m2
Elementary charge
e = 1.6·10-19 C
Diameter of capacitor plates
D = 8 cm
Number of scale units
1 unit ≡ 120 μm
Cunningham constant
A = 0.864
I.
Problems about theory
1.
Stokes frictional force
The video (download from RCL-website, Material, 2.) is showing a falling glass
sphere (ρglass = 2.23 g/cm3, r = 2 mm) in oil (density ρoil = 0.922 g/cm3, viscosity ηoil =
0.09 Ns/m2 of oil made out of sun flowers).
a) Plot a qualitatively correct stroboscope picture of the motion of the glass sphere
and assign all forces acting on that sphere:
ƒ
Which type of all forces are constant or variable starting with the free fall?
ƒ
Explain, why the glass sphere is falling down with constant velocity shortly after letting off.
b)
Study the trajectory of the moving glass sphere by means of video analysis. Show
that the motion of the glass sphere happens under the action of the Stokes frictional
force ( Fs = 6πηoilrv ) and determine all acting forces.
c)
How one may modify the experiment to demonstrate Fs ~ v?
d)
How one may modify the experiment to demonstrate Fs ~ r?
2.
Discrete values
Characteristic of discrete values is that the suited
value may not assume any possible value but only
certain discrete ones. Three examples from every
day life, technique and from mathematics are: staircase (width of steps 20 cm, height of steps 10 cm),
soda machine (Cola 1 €, water 0.8 €, mix drink 1.3
€) in Fig. 1 and series of numbers f(n) = 1/n (n ∈
lN).
Fig. 1: Discrete or continious?
a) Specify which value is discrete in these examples and display the sequence on a
numbered line.
b) What is the difference of the discretization of the staircase with respect to the other
examples?
c)
How one can technically overcome the discretization in case of the staircase?
d) Specify similar examples of every day life, technique and mathematics with continuous values.
4
3.
History of Millikan’s experiment
Millikan’s experiment is at the end of a series of experiments (Improved method to
determine elementary charge from water droplets, laws of electrolysis, determination of specific charge e/me, determination of elementary charge e of oil droplets,
hypothesis about atomic character of electricity, determination of elementary charge
e of water droplets, particles of electricity are called an „electron“) by several physicists (H. A. Wilson, J. J. Thomson, B. Franklin, M. Faraday, R. A. Millikan, G. J.
Stoney, J. S. E. Townsend) since 1750 (1903, 1881, 1909 - 1913, 1897, 1747,
1897, 1833).
a)
Put this information (serious of experiments, physicist, time) in a temporal table.
b) Describe in detail one of these pre-experiments.
4.
Calculation of values in Millikan’s experiment
Systematic measurements gave the following measured values with technical parameters: density of oil ρoil = 1.03 g/cm3, distance between capacitor plates d = 6
mm, viscosity of air ηair = 1.81·10-5 Ns/m2, applied voltage U = 600 V, time interval
for raising trise = 17.4 s and falling tfall = 6.8 s between 5 scale units (1 unit ≡ 120
μm).
a) Calculate the values from all these given parameters for the specific kind of Millikan’s experiment used here in the RCL variant.
5.
Model experiment to Millikan’s experiment
The main components of such a model experiment for Millikan’s experiment are oil,
spheres and pieces of certain mass.
a) Make a sketch of a possible experimental setup for such a model experiment.
b)
Put together in a table the analogies and main differences between the real experiment of Millikan and the model experiment.
6.
Different types of Millikan’s experiment
In the following we study different types of Millikan’s experiment. Buoyancy force
and Cunningham correction are not considered:
a)
What forces are relevant for the motion of the oil droplet? Write down the mathematical dependency between the forces for a constant vector of velocity.
b) To describe the direction of forces and motion during a first (index 1) state of motion
of the oil droplet by scalars we introduce a vertically upwards directed y-axes. Assignments for oil droplet: Charge Q < 0, velocity v1, radius r, density ρÖl. Assignments for capacitor: distance between plates d, voltage U1, viskosity of air ηair:
Write down the scalar equation for the balance of forces. What conditions for U1
and v1 must be fulfilled for the states of motion „floating“, „raising“ and „falling“?
5
c)
One needs always two states of motion (1 and 2) of the oil droplet to determine the
charge Q of the oil droplet: why?
d) In textbooks one can find three types of Millikan’s experiment with state of motion
1/state of motion 2 of the oil droplet:
• Floating/Falling without E-field
• Raising with E-field/Falling without E-field (used in RCL)
• Raising with E-field/Falling with E-field in opposite direction and the same electric
field strength
Derive for at least one type the formula to determinate the charge Q of the oil droplet.
e) The most general type of Millikan’s experiment is „Raising with E-field/Falling with
E-field“ at different electric field strengths:
Derive the formula for Q. Proof the formula by the choosen special case in d).
f)
How all formulas for Q must be modified, if one considers either buoyancy and the
Cunningham correction? Show why the force of buoyancy can be indeed neglected.
g) Why are all types of the experiment belonging to the floating case less suited to determine Q than all others?
7.
Selection of oil droplets
The histogram (Fig. 2) shows the distribution of relative charges k = Q/e of about n =
230 oil droplets of the Millikan experiment.
a) For small values of k the distribution shows
discrete values: why is this discrete pattern
smeared out for larger values of k?
b)
Search for a relation between charge and
velocity of oil droplets looking up data material in the RCL-website, Analysis, 1. respective theoretical considerations or using
the data material itself.
c)
What is the proper recommendation for the
performance of Millikan’s experiment according to the results and experiences of
problem a) and b).
8.
Acceleration phase of oil droplets
Fig. 2: Histogram for absolut frequency h of measured oil droplets
versus relativ charge Q/e.
Under the action of gravitational and electric forces (buoyancy will be neglected here) oil droplets will be accelerated in air to a final constant velocity vfall (during falling
without an electric field) or velocity vrise (during raising with an electric field):
6
a)
Explain the existence of such final stationary velocities.
b)
Estimate the time interval during acceleration of an oil droplet (ρoil = 1.03 g/cm3, r ≈
0.8 μm, Q = 3e) till it reaches the final constant velocity during falling and raising
motion (U = 600 V, d = 6 mm, ηair = 1.81·10-5 Ns/m2).
Hint: suppose a constant force in Newton’s axiom acting on the oil droplet.
c)
The velocity of a spherical body (radius r, density ρ) during falling in a medium (viscosity η) with initial velocity v0 = 0 is given by
v(t) =
g
(1 − e −kt )
k
k=
9η
:
2r 2ρ
ƒ
Show that v(t) is a solution of the differential equation mv& (t) = mg − 6πηrv (balance
of forces).
ƒ
Display a graph of v(t) with the given values in b), determine the time interval for
acceleration and explain the difference to result found in b).
d) Do we have to consider this acceleration time interval for the determination of Q?
9.
Cunningham correction
In Millikan’s experiment we have to use a corrected viscosity for the motion of a
spherical body (radius r) through a medium (dynamical viscosity η, average free
path length λ, Cunningham constant A)
ηcorr (r) =
η
Aλ
1+
r
a) What is the average free path length λ in gases? What is the relation between viscosity of a gas and the average free path length λ in that gas?
b) Under which real conditions do we have to use instead dynamical viscosity η the
corrected one ηcorr for the motion of a body with radius r through a medium?
ƒ
Applying this correction do we increase or decrease the Stokes frictional force?
c)
Plot ηcorr,air/ηair for air at normal conditions (A = 0.864, ηair = 1.81·10-5 Ns/m2, λair =
68 nm) as a function of radius r of oil droplets. Looking at the formula for ηcorr or at
this graph: what is the influence of ηair?
10. R. A. Millikan
Millikan (Fig. 3) was a interesting and many-sided
person:
a) Search for information about Millikan as a researcher, as a private person or as a teacher.
Fig. 3: Robert Andrews
Millikan (1868 - 1953), (VII.6.)
7
II. Problems about experimental setup
1.
Experimental setup of the RCL “Millikan’s Experiment”
Fig. 4 shows numbered parts of the RCL “Millikan’s Experiment”.
Fig. 4: Pictures of experimental setup (left), control panel (middle),
webcam picture and stop watch (right).
a) Set up a table with three columns such as number, assignment and its function and
fill it up looking at the pictures.
2.
Observation of oil droplets
a) What is the difference between bright field
and dark field illumination? Which one is
used in the RCL?
b) If light is scattered by oil droplets which kind
of scattering exists in the RCL - Rayleigh or
Mie scattering? Give a typical example for
both kinds of scattering.
c)
Fig. 5: Webcam and microscope to
observe the oil droplets in the RCL.
The webcam has a teleobjective with 13.5
cm focal length, the focal length of the microscope objective is 5 cm, the focal length of the ocular is 2.5 cm (Fig. 5):
Make a sketch of the optical rays between oil droplets and the CCD chip of the
camera.
d) How can one measure distances in the μm-range with a microscope?
e) In the RCL a bulb of the illumination unit was replaced by a white LED. At the real
Millikan experiment a glass container with copper chloride water solution was positioned between arc lamp and capacitor: What do one want to avoid in both cases?
Without both improvements one would expect perturbing influences on the determination of charge Q. Which kind of perturbations?
f)
How many pictures can be transmitted by ISDN (8 kB/s) or by DSL 1000 (128 kB/s)
using a compression factor of 20 in JPEG format with a resolution of 320 x 240 pixels and 24 bit colour depth?
8
3.
Generation of oil droplets
An airbrush compressor is used to blow a pulsed stream of air to the atomizer (Fig.
6), controlled by mouse click to open a magnetic valve.
a) Quote and describe an experiment which demonstrates, that the static pressure in
flow of liquid or gas is as smaller as the large becomes the velocity of the flowing
liquid or gas.
b) Explain qualitatively how one can succeed in atomizing the oil
into small droplets.
c)
Which velocity of air vair (ρair = 1.3 kg/m3) must be produced by
the airbrush compressor at the glass nozzle (2 cm as distance to
the inlet for oil droplets of the capacitor) to atomize the oil (ρoil =
1.03 g/cm3).
d) Why one uses oil of high vacuum pumping systems in the Millikan
experiment for the oil droplet?
4.
Fig. 6: Atomizer
in the RCL.
Control of the electric field in the capacitor
Fig. 7 shows the module and Fig. 8 the
circuit to control the electric field in the
capacitor (distance between plates d = 6
mm, diameter of plates D = 8 cm). The
high voltage power supply (output voltage
0 < Vout < 1 kV, maximum current IA,max =
1 mA) is controlled by the PC via DA converter:
a) In which range one can vary the electric
field strength?
Fig. 7: Module to control the electric field
in the capacitor and to steer the blow in of
oil droplets into the capacitor.
b) How quickly one can recharge the capacitor?
c)
For what do we need in that circuit the relay?
Fig. 8: Wiring circuit to steer the electric field in the capacitor.
9
5.
Millikan’s original experimental setup
Fig. 9 shows a cut through Millikan’s original apparatus for the determination of the
elementary charge:
Fig. 9: Schematic drawing of Millikan’s original apparatus (VII.5., p 103).
a) Assign in a table the numbers in the sketch to the names of all components including the function of each of the components: arc lamp, toggle switch, pump connections, manometer, water cell, battery, atomizer, chamber, telescope, air condenser,
copper chloride cell, x-ray tube, oil tank, isolator, voltmeter, connection bar.
Functions: measurement of voltage of capacitor, reverse voltage of capacitor,
measurement of pressure in chambers, production of light for illumination, variation
of charge of oil droplets, generation of oil droplets, keeping constant temperature,
connection to the pump to vary pressure of chamber, observation of oil droplets,
absorption of heat radiation, generation of high voltage, generation of a homogeneous electric field, charge and discharge capacitor.
b) Describe in a table the functions for the possible positions of
component 10 (Fig. 10) to the connections of component 18:
Position of
component 10
Circuit connection of
component 18
Function
Fig. 10: Enlargement of component 10.
10
III. Problems about measurements and data analysis
1.
Performing Millikan’s experiment
a) Establish a spreadsheet table for the following measured values U, trise and tfall and
the calculated ones vrise, vfall, r, ηcorr and Q. Control the correctness of the table calculating one final result as an example.
b) Write a precisely formulated manual to perform the measurements with this RCL
experiment.
c)
How big is this part of the capacitor observable by the webcam picture?
d) After applying the electric field some of the oil droplets are rising, but some of them
are falling: what is the reason? How one can separate between both cases during
falling?
2.
Evaluation of Millikan’s experiment
a) Quote two kinds of graphically presenting the measured values in the Millikan experiment. Which are the advantages respective disadvantages of each presentation?
b) If the accumulations around Q = ke are clearly separated in the histogram then one
can determine the elementary charge (nk is the number of Q values in class k):
n1
e=
n2
∑ Qi + ∑
i=1
Qj
+ ...
2
n1 + n2 + ...
j =1
Explain the formula. Determine the elementary charge from own measured values
or as an exercise with data from website.
c)
A function depending on n variables is given in the following form
f(x1,x 2,...,xn ) = C ⋅ x1a1 ⋅ x 2a2 ⋅ ... ⋅ xnan . If each variable xi deviates by Δxi than the total
deviation Δf is given by
2
2
2
⎛ Δx ⎞ ⎛ Δx ⎞
⎛ Δx ⎞
Δf = f(x1,x 2 ,...,xn ) ⋅ ⎜ a1 1 ⎟ + ⎜ a1 2 ⎟ + ... + ⎜ an n ⎟ .
x1 ⎠ ⎝
x2 ⎠
xn ⎠
⎝
⎝
In the following the absolute error ΔQ and relative error ΔQ/Q of the elementary
charge shall be estimated in the specific kind of experimenting of the RCL experiment here. Since the raising and falling velocities are almost equal one can put for
simplicity vrise = vfall = v in the formula for the determination of charge Q. One must
put v = s/t (trise = tfall = t), where either the falling and raising distances s or the falling and raising time intervals t possess errors.
ƒ
Transform the formula for the definition of Q in the following form
f(x1,x 2,...,x n ) = C ⋅ x1a1 ⋅ x 2a2 ⋅ ... ⋅ x nan .
11
ƒ
Establish a table with respect to the determination of charge Q and determine
ΔQ/Q. Some of the deviations Δxi must be estimated or taken from manuals.
Value and its
abbreviation
Unit
xi
12
Δxi
Δx i
xi
⎛ Δx i ⎞
⎜ ai
⎟
xi ⎠
⎝
2
IV. Solutions to problems I
1.
Stokes frictional force
a) See Fig.11.
ƒ
The only changing term along this distance is the Stokes
frictional force Fs.
ƒ
At the time when the glass sphere is released its velocity
is v = 0. The sphere starts falling, because the gravitational force Fg is larger than the force due to buoyancy
Fb. The sphere will be accelerated till the increasing
Stokes frictional force – increasing with increasing velocity during falling – is reaching a dynamical balance of
forces. The resulting total force F on sphere is now zero
and the ball is falling with constant velocity.
b) If the formula for the Stokes frictional force is correct,
then the experimentally determined velocity during falling
vexp is like the theoretically determined velocity during falling vtheo: with Δs = 10 cm and Δt = 0.833 s from the
video we get vexp = 12 cm/s. For vtheo we get
Fg = Fs + Fb ⇒ v theo =
2r 2g(ρglass − ρoil )
9ηoil
= 12.6
cm
.
s
Inserting all technical parameters we get for the forces
Fs = 0.4 mN, Fg = 0.733 mN and Fb = 0.33 mN.
c)
Fig. 11: Stroboscopic view
of a falling sphere of glass.
One must use a set of balls with identical radius r but different density ρ of ball material (different mass m of ball). We measure the radius r and the constant velocity
vexp during falling. We can determine the frictional force Fs from the dynamical balance of forces (for vexp = constant) according to
Fs = mg − moilg = (m − moil )g =
4 3
πr g(ρ − ρoil )
3
Alternatively one can use different kinds of oils with different oil densities. This is
problematic, because each kind of oil with different density possesses also different
viscosity. Unless one is looking for oils with different density but the same viscosity.
d) To show Fs ~ r one has to vary the sphere radius r and keep the velocity constant.
But increasing r k-times at constant density ρ of balls will increase velocity v quadratically.
v = k2 ⋅
2r 2 g(ρ − ρoil )
9ηoil
So to keep v = constant the difference in densities must be put k2 smaller if the radius is increasing by a factor of k.
13
2.
Discrete values
a) Staircase, height of steps with respect to horizontal: 20 cm, 40 cm, …
Soda machine, costs for beverages: 0.8 €, 1.0 €, 1.3 €, …
series 1/n, values for this function: 1, 0.5, 0.25, … .
b) The difference between succeeding discrete steps are constant.
c)
Build up an inclined plane.
d) Every day life: time interval for taking a shower. Technique: velocity of a car. Mathematics: values for the function f(x) = x with x ∈ lR.
3.
History of Millikan’s experiment
a)
Year
Physicist
Development or contribution
1747
B. Franklin
Hypothesis about atomic nature of electricity
1833
M. Faraday
Laws of electrolysis
1881
G. J. Stoney
Particles of electricity are called electron
1897
J. J. Thomson
Determination of specific charge e/me
1897
J. S. E. Townsend
Determination of e by water droplets
1903
H. A. Wilson
Improvement of last technique
1909 - 1913
R. A. Millikan
Determination of e by oil droplets
b) Links about the history of the electrons (checked 01-15-2010):
• http://www.aip.org/history/electron
• http://www.egglescliffe.org.uk/physics/particles/electron/electron.html
• http://www-istp.gsfc.nasa.gov/Education/whelect.html
• http://www.historyofelectronics.com
4.
Calculation of values in Millikan’s experiment
a) vfall = 8.82·10-5 m/s
r = 0.84 μm
Q = 3.14·10-19 C
moil = 2.55·10-15 kg
Fe = 3.14·10-14 N
Fs,rise = 0.923·10-14 N.
vrise = 3.45·10-5 m/s
ηcorr,air = 1.69·10-5 Ns/m2
E = 100000 V/m
Fg = 2.5·10-14 N
Fs,fall = 2.3·10-14 N
14
5.
Model experiment to Millikan’s experiment
a) This model experiment is a variation of the Atwood’s
machine with weight-force of the sphere Fg,m, with
weight-force of mass Fg,M and Stokes frictional force Fs
(Fig. 12). In the dynamic balance of forces
r
r
r r
Fg,m + Fg,M + Fs = 0 .
The difference between the two weight-forces is
equivalent to the Stokes frictional force.
b)
6.
Millikan’s experiment
Model experiment
Weight-force of oil droplet
Weight-force of sphere
Electric force
Weight-force of mass M
Medium air
Medium oil
Buoyancy can be neglected
Buoyancy must be considered
Raising (U > Ufloat), floating
(U = Ufloat), falling (U < Ufloat)
Raising (M > m), floating (M =
m), falling (M < m)
Stokes frictional force Fs with
Cunningham correction
Stokes frictional force Fs
(Reynolds number Re << 1)
Fig. 12: Model experiment to Millikan’s
experiment.
Different types of Millikan’s experiment
r
r
a) On the oil droplet are acting three forces: Weight force Fg , electric force Fe and
r r
r
Stokes frictional force Fs . At constant velocity the resulting force F = 0 and therer r r r
fore Fg + Fe + Fs = 0 .
Fig. 13: Balance of forces for different states of
motion of oil droplet.
b) Fig. 13 shows the direction and
r balance of forces for a floating, falling and raising
oil droplet. FG < 0, because FG is opposite directed to the y-axis. The scalar equation
U
U
4
4
− πr 3ρoilg − Q 1 − 6πηair rv1 = 0 ⇔ πr 3ρoilg + Q 1 + 6πηair rv1 = 0
3
d
3
d
15
describes all states of motions of the oil droplet correctly, if we consider some conditions for U1 and v1:
• Floating oil droplet: It´s v1 = 0 and Fe > 0. We can define the sign of U1 with the
arithmetic operator before QU1/d (Q < 0):
4πr 3ρoilgd
U
4 3
πr ρoilg + Q 1 = 0 ⇔ U1 = Ufloat = −
> 0.
3
d
3Q
Therefore for a electric force in y-direction we have U1 > 0. The voltage Ufloat > 0,
when the oil droplet is floating, we call floating voltage.
• Raising oil droplet: It´s v1 > 0 and U1 > Ufloat > 0.
• Falling oil droplet: It´s v1 < 0 and U1 < Ufloat. We can differentiate between three
cases:
Falling with electric force in opposite to direction of motion (0 < U1 < Ufloat)
Falling without electric force (U1 = 0)
Falling with electric force in direction of motion (U1 < 0).
c)
The radius r of the specific oil droplet cannot be measured. In the respective equations either radius r or both radius r and charge Q are not known. To determine the
charge Q we need a system of equations with two equations and to unknown values r and Q.
d) As an example we derive the formula for the the type of Millikan’s experiment used
in the RCL. During the oil droplet is raising we have
U
4
− πr 3ρoilg − Q 1 − 6πηair rv1 = 0 .
3
d
During the oil droplet is falling we have
9η v
4 3
4 πr 3ρoilg
− πr ρoilg − 6πηair rv 2 = 0 ⇔ 6πηair r = −
⇔ r 2 = − air 2 .
3
3 v2
2ρoilg
Inset of this equation in the first equation delivers
U 4 πr 3ρoilg
v
U
4
4
− πr 3ρoilg − Q 1 +
v1 = 0 ⇔ πr 3ρoilg( 1 − 1) = Q 1
3
d 3 v2
3
v2
d
9η v
v
U
4 9η v
⇔ − π ⋅ air 2 ⋅ − air 2 ⋅ ρoilg( 1 − 1) = Q 1
3
2ρoilg
2ρoilg
v2
d
⇔Q=−
η v
18πd
− air 2 ⋅ (v1 − v 2 )
U1
2ρoilg
Because of U1 > 0 and v2 < 0 the charge Q < 0.
e) For raising (state of motion 1) and falling (state of motion 2) we have
U
4 3
πr ρoilg + Q 1 + 6πηair rv1 = 0
3
d
U
4 3
πr ρoilg + Q 2 + 6πηair rv 2 = 0 .
3
d
Subtraction of the second equation from the first equation delivers
16
Q(U1 − U2 )
Q
(U1 − U2 ) + 6πηair r(v1 − v 2 ) = 0 ⇔ r = −
.
d
6πηair d(v1 − v 2 )
Inset of this equation in the first equation for raising of the oil droplet delivers
Q3 (U1 − U2 )3
U Q(U1 − U2 )
4
v1 = 0 .
− πρoilg
+Q 1 −
3
3
3
(6πηair d) (v1 − v 2 )
d d(v1 − v 2 )
After division by Q ≠ 0, multiplication by d(v1-v2) and transformations we get
3π2 (6ηair )3 d2 (v1 − v 2 )2 (U2 v1 − v 2U1 )
ηair 3 (U2 v1 − v 2U1 )
⋅ (v1 − v 2 ) .
Q =
⇔ Q = −18πd
4π(U1 − U2 )3 ρoilg
2ρoilg(U1 − U2 )3
2
• Because of U1 > 0 and U2 < 0 is (U1 – U2)3 > 0
• Because of v2 < 0 and U1 > 0 is –v2U1 > 0
• Because of v1 > 0 and v2 < 0 is v1 – v2 > 0
• Because of v1 > 0 is U2v1 > 0 for 0 < U2 < Ufloat and U2v1 < 0 for U2 < 0
Therefore the radicant is positive. Because of Q < 0 we choose the negative sign
during extraction.
We proof the formula with the simplest type of Millikan’s experiment containing the
states of motion „Floating/Falling without E-field“. For floating is v1 = 0 and U1 = Ufloat, for falling is U2 = 0. Therefore we get
Q=
18πdηair v 2
Ufloat
−ηair v 2
.
2ρoilg
Because of v2 < 0 the radicant is positive and Q < 0. The formula is the same as in
d) or in textbooks.
f)
Since ρair is about a factor 1000 smaller than ρoil:
Fg − Fb =
4 3
4
πr g(ρoil − ρair ) ≈ πr 3 gρoil = Fg .
3
3
The density of oil must be replaced by ρoil – ρair, because in the balance of forces Fg
must be replaced by Fg – Fb. In addition, the viscosity of air must be corrected too,
i.e. the frictional force Fs must contain ηcorr,air.
g) The velocity v is hardly influenced by changes of the acceleration voltage U: For a
typical oil droplet (Q = 2e and r = 1 μm), i.e. the floating condition is hardly to be realized and to be watched.
QU 4 3
− πr ρoilg
4 3
QU
d
3
πr ρoilg =
− 6πηair rv ⇔ v(U) =
3
d
6πηair r
μm
dv
Q
=
= 0.156 s
dU 6πηair rd
V
A change in voltage by 10 V causes a change in velocity of 1.5 μm/sec. The oil
droplet will need about 8 sec moving slightly by 0.1 scale units (0.1 scale unit = 0.1
17
x 120 μm = 12 μm), i.e. the floating condition is hardly to be realized and to be
watched.
Brownian motion: Because of the thermal motion of air molecules the floating oil
droplet is performing a zig-zag motion by the collisions with air molecules. If s is the
distance between two changes in direction than Δx may be its projection in an arbitrary direction: for the average of squared projections
Δx 2 =
kTτ
.
3πηr
For a temperature T = 300 K, a time interval of observation τ = 8 s between two positions (viscosity ηair = 1.81·10-5 Ns/m2 and radius r = 0.8 μm of oil droplet) we get
Δx 2 = 15 μm . The zig-zag motion due to Brownian motion is several times larger
than the dimension of this oil droplet.
7.
Selection of oil droplets
a) Oil droplets with larger charges are
moving much faster. Therefore,
the error in determining the velocity and therefore in charge Q is increasing.
b) The charging of oil droplets by friction is a statistical process similar
to the process of charge separations of particles in clouds. Therefore, there is no deterministic condition given for that process. Due
to empirical data oil droplets,
which are closer together than
0.45 μm, will be charged negatively: in addition one supposes
empirically that the amount of
charge Q is increasing with increasing radius of oil droplets. This hypothesis is confirmed by the fact that
the capacity
No.
Radius
r in
10-7 m
Velocity
vfall in
10-5 m/s
Velocity
vrise in
10-5 m/s
Charge
Q in
10-19 C
1
7.86
7.66
5.30
3.05
2
5.34
3.53
7.16
1.60
3
8.44
8.83
3.50
3.14
4
8.51
8.97
3.26
3.15
5
10.6
13.9
20.1
11.19
6
7.47
6.92
7.83
3.28
7
7.23
6.47
8.12
3.12
8
6.26
4.86
12.8
3.19
9
9.68
11.6
20.1
9.42
10
6.99
6.06
9.07
3.11
12
10
8
Csphere = 4πε0r
6
is increasing with radius r.
4
Using Fig. 14 we can model in a
simple linear model the negative
charge of an oil droplet (for r > 0.45
μm):
2
Q(r) = 1.823 ⋅ 10 −12
0
0
2
4
6
8
10
12
-7
Radius r in 10
mm
Öltröpfchenradius
r / 10-7
Fig. 14: Measured charge Q of oil
droplets versus radius r of oil droplet.
C
(r − 0.45 μm) .
m
18
We have to distinguish between raising and falling of an oil droplet. The velocity of
falling vfall depends on the radius r only.
12
2ρ g
4 3
πr ρoilg = 6πηair rv fall ⇔ v fall = oil r 2
3
9ηair
10
8
An oil droplet with double radius is falling
four times as fast. In general, larger oil
droplets possess on average also larger
charge Q, therefore, all those quickly falling droplets possess larger charge.
6
4
2
0
0
According to empirical data (Fig. 15) the
charge Q depends also on the velocity of
raising vrise. What do we expect for
vrise(Q,r) in such a simple linear model?
5
10
15
20
St eiggeschwindigkeit
in um/s
Raising
velocity vrisevs in
μm/s
Fig. 15: Measured charge Q of oil
droplets versus raising velocity vrise.
U
4 3
πr ρoilg = Q rise − 6πηair rv rise ⇔ v rise =
3
d
Q
Urise 4 3
− πr ρoilg
d
3
6πηair r
and
Q(r) = 1.823 ⋅ 10 −12
C
m
(r − 0.45 μm) ⇔ r(Q) = 5.485 ⋅ 1011
⋅ Q + 0.45 μm .
m
C
If we insert both dependencies vrise(Q,r) and r(Q) into the definition equation of the
raising velocity vrise we get as graphical dependence vrise(r) in Fig. 16 and vrise(Q) in
Fig. 17:
vrise in m/s
vrise in m/s
Fig. 16: Raising velocity vrise versus
radius r of oil droplet.
Fig. 17: Raising velocity vrise versus
charge Q of oil droplet.
Because the gravitational force Fg is increasing with radius r larger than the electric
force Fe with Q the raising velocity vrise is decreasing again for oil droplets with radius r > 1 μm or Q ~ 6e.
c)
Since only a few oil droplets in the Millikan experiment are larger than r > 1 μm or
possess a charge Q ≥ 6e it is suggested to select only smaller oil droplets to guarantee a possibly accurate determination of charges.
19
25
8.
Acceleration phase of oil droplets
a) See I.1a.
b) In the very beginning of the falling process the resultant acceleration is a(t = 0) = g.
The velocity of falling is
2ρ g
4 3
μm
πr ρoilg = 6πηair rv fall ⇔ v fall = oil r 2 = 0.8
.
3
9ηair
s
The time interval of the acceleration phase is
Δt fall =
v fall
= 8 μs .
g
During raising at the beginning the balance of forces is F = Fe – Fg using the velocity of raising vrise from I.7b:
Δt rise
c)
ƒ
4
F πr 3ρoil
v rise
2r 2ρoil
=
= 3
=
= 8.1 μs .
a
6πηair rF
9ηair
The evolution of the falling velocity v(t) is
6πηrv
9η v
= g − 2 = g − kv
4
2r ρK
ρK πr 3
3
g
= g − k (1 − e−kt )
k
& = mg − 6πηrv ⇔ v(t)
& = g−
mv(t)
g
− ( −k)e−kt
k
ƒ
Using all measured values we get for k =
1.23·105 s-1. The time interval Δtfall, till the
falling velocity is constant, is ≈ 40 μs (Fig.
18). The time interval, till the falling velocity is about 63 % of the final velocity v∞ =
vfall, is about 1/k = 8 μs considering
vfall in μm/s
1 g
(1 − ) ⋅ = 0.63 ⋅ v ∞ .
e k
Comparing both results Δtfall ~ 8 μs, Δtfall
~ 40 μs from b) the time interval for acFig. 18: Falling velocity vfall versus time t.
celeration is too large in c) than in b), because in c) we used a too high constant gravitational force during the whole phase
of acceleration.
d) Therefore the acceleration times in the falling and raising phases can be neglected
with respect to the reaction times of the user, if an oil droplet changes its direction
under the action of an applied voltage.
20
9.
Cunningham correction
a) In a gas the number of collisions per second of one particle (gas molecule) is proportional to the number density n, to the radius r of particle and to the average velocity of particles v (see VII.1.):
Z = 4 2πnr 2 v
At ambient conditions (n = NA/Vmol, r ≈ 10-10 m und v ≈ 103 m/s) the number of collisions per second of a gas particle with other ones is ~ 48⋅106. Therefore, the mean
free path length λ of one particle between two collisions is
λ=
v
= 210 nm
Z
The viscosity of a gas is proportional to the mean free path length:
η=
1
nmvλ .
3
b) If the radius r is of order of the mean free
path length λ one has to correct the viscosity. In our case λ ~ 0.068 μm and r ~ 0.1
μm to 1 μm.
ƒ
Stokes force decreases.
c)
The (empirical found) Cunningham correction is ηcorr < η if the following inequality
holds
Aλ
> 0 ⇒ ηcorr < η
r
ηcorr,air/ηair
Fig. 19: Factor of correction for viscosity of air versus radius r of oil droplet.
with A constant. Fig. 19 shows the dependence between the correction and radius
of oil droplet. The Stokes force will be smaller, since the oil droplets may fall down
easier between the air molecules. Viscosity η is the limes for infinitely large sphere.
10. R. A. Millikan
a) Links to the life of Millikan:
• http://nobelprize.org/nobel_prizes/physics/laureates/1923/millikan-bio.html
• http://millikan.kegli.net/Millikan.htm
• http://scienzapertutti.lnf.infn.it/biografie/millikan-bio_fra.html
21
V. Solutions to problems II
1.
Experimental setup of RCL “Millikan’s Experiment”
a)
2.
No.
Assignment
Function
1
Microscope
To magnify oil droplets
2
Capacitor
To produce an electric field
3
Stepper motor
To focus on one oil droplet
4
Oil atomizer
To generate oil droplets
5
Light source
To illuminate oil droplets from aside (dark field)
6
Button
To change electric force
7
Button
To blow droplets into capacitor
8
Buttons
To shift focal plane inside capacitor
9
Buttons
To displace horizontally the section of picture
10
Button
To turn on or off the electric field
11
Microscope picture
Enlarged representation of oil droplets
12
Oil droplet
Object under investigation
13
Ocular micrometer
To measure distances in the range of μm
14
Stop watch
To measure raising time
15
Stop watch
To measure falling time
Observation of oil droplets
a) In bright field illumination light transmitted through an object is collected by the microscope objective. In dark field illumination only the light scattered by an object is
collected by the microscope objective.
b) We are speaking of Rayleigh scattering, if the dimension d of a scattering object is
smaller than the wavelength λ of scattered light (e.g. light scattering by gas molecules, this fact is explaining the blue colour of sky, the red colour of sun rise and
sun dawn). We are speaking of Mie scattering if the dimension d of an object is of
the same order or larger than the wavelength λ (e.g. scattering by aerosols, by fog
(0.01 mm – 0.1 mm) or by rain drops (0.1 mm – 5 mm). In the Millikan experiment
the diameter of oil droplets (0.1 μm < 2r < 1 μm) is of the same order as the wavelength (0.4 μm < λ < 0.8 μm) which means this scattering is between the one of
Rayleigh or of Mie.
c)
Optical ray diagram of a microscope and of a telescope (functions).
d) We use an ocular micrometer to measure
distances in the microscope, which we
calibrate by an objective micrometer in
the following way (Fig. 20):
• Insert ocular micrometer (transparent
plate with scale divisions, low part of
22
Fig. 20: Ocular micrometer.
Fig. 19) at the focal plane of objective respective at the object plane of the ocular: focussing by means of the ocular lens.
• Put object micrometer (scale division of 1 mm for 100 parts i.e. 1 part = 1 μm,
upper part of Fig. 20) on object carrier of microscope, focussing by rough and
fine drive, than rotate ocular micrometer till it is parallel to objective micrometer.
• Count scale units of both objective and ocular micrometer over a maximum wide
coinciding distance s. In Fig. 19 m = 20 scale units of the object micrometer
represents n = 90 scale units of the ocular micrometer.
• Length x of one scale unit of ocular micrometer:
s = 20 Skt ⋅
10 μm
20
μm
μm
= 90 Skt ⋅ x ⇔ x =
= 2,222
scale unit
9 scale unit
scale unit
e) The heat transfer from the light source by heat radiation to the air in the capacitor
will be prevented by the absorption of the heat radiation by copper chloride. Consequence of this heat transfer is convection in the air inside capacitor, which may
cause a side drift motion of oil droplets and therefore additional vertical forces on
the oil droplets. This side drift motion is influencing the vertical motion of the oil
droplet. In that case for a motion with friction the principle of independency is not
valid any more.
But there is not influence of this side drift motion on the measurement of falling –
and raising velocity, because one is measuring only the vertical (i.e. projected) distance of the motion of a droplet in the ocular micrometer.
f)
Transport of data:
x ⋅ 320 ⋅ 240 ⋅ 3 B ⋅
1
= y B/s
20
For y = 8000 we get α = 0.7 pictures/s, for y = 128 000 we get x = 11.1.
3.
Generation of oil droplets
a) The vertical tubes are measuring the static
pressure p of the liquid (p = ρgh, h height). At
the reduced part (cross section) of the horizontal flow tube the static pressure is smaller than
elsewhere: since the total pressure p0 is constant everywhere the dynamical pressure p =
ρv2/2 is larger at that position.
Fig. 21: Static and dynamic pressure in tubes.
b) The principle of an atomizer is based on two mechanisms:
• According to Bernoulli equation rapidly moving air at the
narrow nozzle will produce a lower pressure at the bottom of the vertical tube with respect to atmospheric
pressure p0. The atmospheric pressure, therefore, will
press the liquid into the vertical tube.
• Liquid will be strongly perturbed by the turbulent flow of
air, such that the liquid will decay into small droplets.
23
Fig. 22: Atomizer.
c)
The minimum velocity will be determined for the case that oil has voil = 0 at the upper end of the vertical tube:
p0 − p = ρoilgh
The air-brush compressor will accelerate the air at rest to a velocity vair at the upper
end of the vertical tube. Following Bernoulli
p0 = p +
ρair
v air 2
2
consequently
ρair
v air 2 = ρoilgh ⇔ v air =
2
2ρoilgh
m
.
= 17.6
ρair
s
d) The saturation vapour pressure of high vacuum oil at T = 20° C is 10-4 kPa much
smaller than the one from water (2.34 kPa). Therefore, the mass of oil droplets will
be constant during measurements.
4.
Control of the electric field in the capacitor
a) Under the assumption of an homogeneous electric field and according to E = U/d
the minimum field strength is Emin = 0 V/m and the maximum is Emax ~ 170 000 V/m.
b) The capacity of the plate capacitor is
πD2
= 7.4 pF .
C = ε0
4d
The internal resistor of the voltage source is Ri = 1 kV/1 mA = 1 MΩ, the resistor of
cables is negligible. To charge a capacitor to the voltage V0 via the resistor R is
VC (t) = V0 (1 − e− t / τ )
with characteristic time constant τ = RC = 7.4 μs. The charging time is much
smaller than the reaction time of user. The current at the beginning of the charging
process is U0/R = 1 mA.
c)
If we switch off the voltage at the capacitor for the falling process then a high voltage relay guarantees that the capacitor will be short circuited. As a consequence
the electric force on the oil droplet disappears immediately, the remaining forces
are in a dynamical balance.
24
5.
Millikan’s original experimental setup
a)
No.
Assignment
Function
1
Atomizer
Generation of oil droplets
2
Chamber
Space inside capacitor
3
Pinhole
Inlet for oil droplets at capacitor
4
Air condenser
Production of a homogeneous electric field
5
Isolator
Isolation between plates of capacitor
6
Stripe
Protection of capacitor space
7
Arc lamp
Production of light to illuminate oil droplets
8
Telescope
Magnification of oil droplets
9
Battery
Production of high voltage
10
Toggle switch
Charge and discharge capacitor
11
Water cell
Absorption of heat radiation
12
Copper chloride cell
Absorption of heat radiation
13
Manometer
To measure pressure in chamber
14
Oil tank
To keep constant temperature
15
Pump connection
Pump connection to chamber pressure
16
X-ray tube
Generation of additional charges of oil droplets
17
Voltmeter
To measure voltage at capacitor
18
Connection bar
Reverse charging of capacitor
b)
Position of switch of
10 (1 or 2)
Connection of electrodes of 18 (e.g. 1-2)
Function
1
2
2-3
Discharge capacitor
2
1,2, 3-4
Charge upper plate of capacitor (-)
2
1,3, 2-4
Charge upper plate of capacitor (+)
25
VI. Solutions to problems III
1.
Performing Millikan’s experiment
a) See I.4a.
b) Look up RCL-website, Tasks.
c)
Approximately 18 scale units x 12 scale units = 2.16 mm x 2.46 mm (with 1 scale
unit ≡ 120 μm).
d) At given connection of the capacitor with positively charged upper plate of the capacitor only negatively charged droplets will raise, whereas either negatively or
positively charged oil droplets will fall down. Increasing voltage negatively charged
droplets are falling more slowly than positively charged ones.
2.
Evaluation of Millikan’s experiment
a)
Histogram
Point diagram
Absolute abundance of measurements versus charge Charge of oil droplets versus number of oil
of oil droplets
droplets respective no. of measurement
Scattering around multiple of elementary charge recog- Quantization (point belonging to Q = ke
nizable increase of this scattering with larger Q
with k = 1,2,3) recognizable
Representation is depending on the width of the class
Meaningful representation with already
chosen
very few data points
b) From groups of data with k-multiple of the elementary charge we will get the elementary charge by division with k. From all these values we determine the average.
As a result for the web site we get e = 1.58 · 10-19 C.
c)
According to VII.4:
Q = 36πd
3 3
3
−
ηair 3 s3
36π − 21 − 21
−1
2 2
2
=
g
ρ
U
d
⋅
t
s
η
oil
rise
air
2 3
2ρoilgUrise t
2
Name of unit
Dimension
xi
Δxi
Gravitational acceleration g*
Density of oil ρoil
Voltage during raising Urise
Distance of plates in capacitor d
Time interval t
Distance s
Viscosity ηair
m/s2
kg/m3
V
m
s
m
Ns/m2
9.8094
1030
600
0.006
8
6·10-4
1.81·10-5
0.0001
0.5
10
0.0005
0.2
1.2·10-5
5·10-7
⎛ Δx i ⎞
⎜ ai
⎟
xi ⎠
⎝
1·10-5
4.8·10-4
0.0166
0.083
0.025
0.02
0.027
Σ
2.5·10-11
5.78·10-8
2.75·10-4
6.8·10-3
1.4·10-3
9.0·10-4
1.64·10-3
0.011
Σ
* for Kaiserslautern, Germany
2
Δx i
xi
0.104
Consequently we get ΔQ/Q ≈ 11 % and ΔQ ≈ 0.16·10-19 C. Our value from b) is lying within the error bars 1.6·10-19 C ± 0.16·10-19 C.
26
VII. References
Since this collection of problems is translated from German version the literature is
due to German sources. Similar literature of * is available in English language.
1.
Gobrecht, H. (1974): Bergmann-Schaefer, Lehrbuch der Experimentalphysik, Mechanik-Akustik-Wärme. De Gruyter, Berlin & New York, S. 652. *
2.
Millikan, R. A.: On the elementary electrical charge and the Avogadro constant,
Physical Review 2 (1913) 2, pp 109-143,
http://authors.library.caltech.edu/6438/01/MILpr13b.pdf, 01-07-10.
3.
Millikan, R. A. (1924): The Electron – Its isolation and measurement and the determination of some of its properties. University of Chicago Press, Chicago & London.
4.
Vogel, D.: Die Auswertung des Millikan-Versuches. PhiS 34 (1996) 3, S. 110-114.
5.
Wilke, H.-J. (1987): Historische physikalische Versuche. Aulis, Köln, S. 101-106. *
6.
http://physics.hallym.ac.kr/~physics/reference/physicist/frankfurt/gif/phys/millikan.jp
g, 01-07-10.
27