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General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Millikan Oil Drop Experiment Ver.20170911
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 1 / 19
[International Campus Lab]
Millikan Oil Drop Experiment
Determine the charge of an electron by observing the effect of an electric field on a cloud of charged oil droplets.
In 1909, Robert Millikan conducted the oil drop experiment
to determine the charge of an electron. He suspended tiny
charged droplets of oil between two metal electrodes by bal-
ancing downward gravitational force with upward electric
force. The density of the oil was known, so he could deter-
mine the droplets’ masses from their observed radii. Using
the known electric field and the value of gravity and mass, he
determined the charge on oil droplets in mechanical equilibri-
um. By repeating the experiment, he confirmed that the
charges were all multiples of some fundamental value. He
proposed that this value was the charge of a single electron.
The figure 1 shows a simplified scheme of Millikan’s oil drop
experiment. A fine mist of oil droplets is sprayed into the
chamber and some droplets enter the space between the
plates. The droplets become electrically charged by an ioniz-
ing radiation source such as an X-ray.
By applying a potential difference across a parallel pair of
horizontal metal plates, a uniform electric field is created in
the space between them. The electric field can be controlled
by changing the voltage across the plates.
The droplets can be observed through a viewing scope with
illumination by the bright light of a lamp. The scope has reticle
marks spaced to 0.1mm, so the velocity of the falling or rising
drop can be calculated.
Fig 1 (a) Simplified scheme of the Millikan oil drop experiment. (b) Forces on a falling drop. (plates not charged) (c) Forces on a rising drop. (plates charged)
Objective
Theory
----------------------------- Reference --------------------------
Young & Freedman, University Physics (14th ed.), Pearson, 2016
23. Electric Potential – Problem 23.81 (p.807)
-----------------------------------------------------------------------------
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Millikan Oil Drop Experiment Ver.20170911
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 2 / 19
Figure 1(b) shows the forces acting on a droplet when it is
falling in air. The velocity of drop has reached its terminal
velocity 𝑣𝑣𝑓𝑓 in a few milliseconds for the droplets used in this
experiment. The frictional force 𝐹𝐹fr = 𝑘𝑘𝑣𝑣 is proportional to the
velocity of the drop, where 𝑘𝑘 is the coefficient of friction be-
tween the air and the drop. Since the gravitational force
𝐹𝐹𝗀𝗀 = 𝑚𝑚𝘨𝘨 and the frictional force 𝐹𝐹fr are equal and opposite,
𝑚𝑚𝘨𝘨 = 𝑘𝑘𝑣𝑣𝑓𝑓 (1)
Figure 1(c) shows the forces acting on the drop when it is
rising under the influence of an electric field. The electric in-
tensity is given by 𝐸𝐸 = 𝑉𝑉 𝑑𝑑⁄ , where 𝑉𝑉 is the potential differ-
ence across the parallel plates separated by a distance 𝑑𝑑. If
the charge on the droplet is 𝑞𝑞 then there is an additional
electric force 𝐹𝐹e = 𝑞𝑞𝐸𝐸 on it. If the sign and magnitude of the
field are such to make the droplet rise, it will quickly acquire a
terminal velocity 𝑣𝑣𝑟𝑟 and the frictional force 𝐹𝐹fr = 𝑘𝑘𝑣𝑣𝑟𝑟 acts on
it. Adding the forces vectorially yields
𝑞𝑞𝐸𝐸 =𝑞𝑞𝑉𝑉𝑑𝑑 = 𝑚𝑚𝘨𝘨 + 𝑘𝑘𝑣𝑣𝑟𝑟 (2)
In both cases, there is also a small buoyant force exerted by
the air on the droplet. Since the density of air is only about
10−3 of that of oil, this force may be neglected.
Eliminating 𝑘𝑘 from equations (1) and (2) and solving for 𝑞𝑞
yields
𝑞𝑞 =𝑚𝑚𝘨𝘨𝑑𝑑𝑉𝑉 �1 +
𝑣𝑣𝑟𝑟𝑣𝑣𝑓𝑓� (3)
To eliminate 𝑚𝑚 from equation (3), we use the expression for
the volume of a sphere
𝑚𝑚 =43𝜋𝜋𝑟𝑟
3𝜌𝜌 (4)
where 𝑟𝑟 is the radius of the droplet, and 𝜌𝜌 is the density of
the oil. Substituting equation (4) into equation (3) yields
𝑞𝑞 =4𝜋𝜋3𝜌𝜌𝑟𝑟3𝘨𝘨𝑑𝑑𝑉𝑉 �1 +
𝑣𝑣𝑟𝑟𝑣𝑣𝑓𝑓� (5)
The velocities 𝑣𝑣𝑓𝑓 and 𝑣𝑣𝑟𝑟 of the equation (5) can be meas-
ured experimentally and only the radius 𝑟𝑟 of the oil droplet
remains unknown. The frictional force 𝐹𝐹fr = 𝑘𝑘𝑣𝑣 in figure 1
actually results from the viscosity between the oil and the air.
The viscous force on a small sphere moving through a vis-
cous fluid is given by
𝐹𝐹 = 6𝜋𝜋𝜋𝜋𝑟𝑟𝑣𝑣 (6)
Equation (6), known as Stokes’ law, is the frictional force
acting on the interface between a fluid and a particle, where
𝜋𝜋 is the viscosity, 𝑟𝑟 is the radius of the spherical object, and
𝑣𝑣 is the flow velocity relative to the object. (For accurate
analysis, Millikan suggested the corrected form of Stokes’ law
with effective viscosity 𝜋𝜋eff = 𝜋𝜋(1 + 𝑏𝑏 𝑝𝑝𝑟𝑟⁄ )−1 , where 𝑏𝑏 is a
constant and 𝑝𝑝 is the atmospheric pressure. We will neglect
this correction factor here.) Now Equation (6) can be substi-
tuted for 𝐹𝐹fr = 𝑘𝑘𝑣𝑣 in equation (1). Substituting equations (4),
(6) into equation (1) yields
43𝜋𝜋𝑟𝑟
3𝜌𝜌𝘨𝘨 = 6𝜋𝜋𝜋𝜋𝑟𝑟𝑣𝑣𝑓𝑓 or 𝑟𝑟 = �9𝜋𝜋𝑣𝑣𝑓𝑓2𝜌𝜌𝘨𝘨 (7)
Substituting equation (7) into equation (5) yields
𝑞𝑞 = 18𝜋𝜋𝑑𝑑𝑉𝑉�𝜋𝜋3𝑣𝑣𝑓𝑓3
2𝜌𝜌𝘨𝘨 �1 +𝑣𝑣𝑟𝑟𝑣𝑣𝑓𝑓� (8)
𝑞𝑞 : Charge carried by the droplet (C)
𝑉𝑉 : Potential difference across the plates (V)
𝑑𝑑 : Separation of the plates of the capacitor (m)
𝜌𝜌 : Density of oil (kg m3⁄ )
𝘨𝘨 : Acceleration of gravity (m s2⁄ )
𝜋𝜋 : Viscosity of air (Pa ∙ s = N ∙ s m2⁄ ) (See appendix.)
𝑣𝑣𝑓𝑓 : Velocity of fall without E-field (m s⁄ )
𝑣𝑣𝑟𝑟 : Velocity of rise without E-field (m s⁄ )
The most precise value of the charge of an electron is
𝑒𝑒 = 1.602176487(40) × 10−19 C.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Millikan Oil Drop Experiment Ver.20170911
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 3 / 19
1. List
Item(s) Qty. Description
Millikan Apparatus
1 Measures the elementary electric charge using a clas-sical method of Millikan.
Power Supply (Power cord included)
1 Supplies voltage for the capacitor plates. Produces regulated DC power up to 50mA in a voltage range 0 to 500 volts.
Power Adapter (for LED lamp)
1 Supplies voltage for the LED lamp.
Patch Cords (High Voltage) (with safety shrouded banana plugs)
2 Connect the power supply to the capacitor plates of the Millikan apparatus.
Patch Cords (with banana plugs)
2 Connect the multimeter to the thermistor of the Millikan apparatus.
Atomizer
1 Sprays oil droplets of density 𝜌𝜌 = 886 kg m3⁄
Multimeter
1 Measures voltage, current, and resistance.
A-shaped Base Support Rod (600mm)
1 2
Provide stable support for experiment set-ups.
Vernier Caliper
1 Measures external or internal diameter of an object with a precision to 0.05mm.
Equipment
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Millikan Oil Drop Experiment Ver.20170911
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 4 / 19
2. Details
(1) Millikan Apparatus
(2) Power Supply
The power supply provides regulated DC power up to
50mA in a voltage range 0 to 500 volts.
(3) Multimeter
The multimeter measures voltage, current, and resistance.
(4) Vernier Caliper
The Vernier caliper measures external, internal diameter or
depth of an object with a precision to 0.05mm.
① 22 mm is to the immediate left of the zero on the vernier
scale. Hence, the main scale reading is 22 mm.
② Look closely for and alignment of the scale lines of the
main scale and vernier scale. In the figure, the aligned (13th)
line corresponds to 0.65 mm (= 0.05 × 13).
③ The final measurement is given by the sum of the two
readings. This gives 22.65 mm (= 22 + 0.65).
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Millikan Oil Drop Experiment Ver.20170911
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 5 / 19
Setup 1. Equipment Setup
(1) Adjust the height of the platform.
Mount the apparatus on two support rods on the A-shaped
base with the viewing scope at a height which permits the
experimenter to sit erect while observing the drops.
(2) Measure the plate separation distance.
Disassemble the chamber by lifting the Housing straight up
and then removing the upper capacitor plate and spacer
plate. Measure the thickness of the spacer (which is equal to
the plate separation distance) with vernier calipers. Be sure
that you are not including the raised rim of the spacer in your
measurement.
𝑑𝑑 = _______________ (m)
(3) Reassemble the chamber.
The flat cut (with black painted hole) of the spacer ring must
face the viewing scope.
Make sure fit the electric discharge terminal into the groove
on the underside of the spacer, and leave no gap between
the spacer and capacitor plates.
Setup
Caution
To prevent electric shock, do not touch the plates inside
the chamber while the power supply is turned on. Prior to
disassembling the chamber, you should be sure to turn
off the power supply and rotate the plate charging knob
to the Plate Grounded (middle) position.
Caution
Be careful not to lose any parts of the chamber assem-
bly, especially the droplet hole cover.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Millikan Oil Drop Experiment Ver.20170911
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 6 / 19
(4) Turn on the LED lamp.
Connect the 12V DC adaptor to the lamp power jack.
(5) Focus the scope and adjust the position of the lamp.
Remove the droplet hole cover. Unscrew the focusing wire
from its storage place on the platform and carefully insert it
into the hole in the center of the top capacitor plate.
Bring the reticle into focus by turning the reticle focus ring.
View the focusing wire through the viewing scope and bring
the wire into sharp focus by turning the droplet focusing ring.
Adjust the position of the lamp by using two lamp position
knobs.
The light is best focused when the right edge of the wire is
brightest (in highest contrast compared to the center of the
wire).
Return the focusing wire to its storage location on the plat-
form.
Complete the reassembly of the chamber by placing the
droplet hole cover on the upper plate and the lid on the hous-
ing. (If you do not cover the droplet hole of the plate, the hole
could be clogged with oil.)
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Millikan Oil Drop Experiment Ver.20170911
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 7 / 19
(6) Realign the optical system while observing the droplets.
Move the air vent lever to the OPEN positon. Place the noz-
zle of the atomizer into the hole on the lid of the chamber and
squeeze the atomizer bulb with quick squeeze. While viewing
a shower of drops through viewing scope, realign the optical
system.
(7) Determine the temperature of the chamber and calculate
the viscosity of the air.
Connect the multimeter to the thermistor connectors and
measure the resistance of the thermistor. Refer to the Suth-
erland’s Formula and the Thermistor Resistance Table (see
appendix) to find the viscosity of air.
𝜋𝜋 = _______________ (Pa ∙ s)
(8) Connect the power supply to the plate voltage connectors.
(9) Set the output voltage of the power supply.
Prior to turning on the power supply, set the plate charging
knob to Plate Grounded (middle) position and rotate voltage
adjustment knob fully counterclockwise (𝑉𝑉 = 0 V).
Turn on the power and increase SLOWLY the output volt-
age to 300 V.
Caution
To prevent electric shock, use ONLY safety patch cords
with shrouded banana plug.
Do not apply voltage to the thermistor connectors.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Millikan Oil Drop Experiment Ver.20170911
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 8 / 19
Setup 2. Stopwatch Software
(1) Run Watchy software.
(2) How to use the Watchy.
[Start] or [F5] : Start the clock.
[Stop] or [F6] : Stop the clock.
[Reset] or [F7] : Reset the stopwatch to zero. Clear the log.
[Lap] or [F10] : Add a lap time without stopping the clock.
[Split] or [F11] : Add a split time without stopping the clock.
[Split] records overall time at any given point, whereas [Lap]
records elapsed time between splits.
Step 1. Setting the Plates Voltage
(1) Set the potential difference across the parallel plates.
Set the output voltage of the power supply to 300 V or any
desired value. (Do not exceed 400 V.)
𝑉𝑉 = _______________ (V)
(2) Rotate the plate charging knob of the power supply to the
Plate Grounded (middle) position.
The plate charging knob changes the direction of the electric
field between the plates. We will introduce the droplets into
the chamber with no electric field by setting the knob to the
Plate Grounded position.
Procedure
Note
It is recommended you do not use any air conditioner or
fan while performing your experiment. The airflow outside
the chamber could affect the motion of oil droplets. The
high velocity stream of air creates a region of lower pres-
sure above the chamber lid, compared with standard
atmospheric pressure inside the chamber. This pressure
difference results in a net force pushing droplets up.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Millikan Oil Drop Experiment Ver.20170911
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 9 / 19
Step 2. Introducing Droplets into the Chamber
(1) Move the air vent lever to OPEN position.
Move the air vent lever to the OPEN position to allow air to
escape from the chamber during the introduction of droplets
into the chamber.
(2) Introduce the droplets into the chamber.
(3) When you see a shower of drops, move the air vent lever
to the CLOSE position.
Step 3. Charging the Droplets
The apparatus uses the piezo igniter to charge droplets. An
applied mechanical stress on piezoelectric ceramic in the
igniter generates a high voltage. The voltage produces an
electric field in the gap between the end of the connected
wire and the lower plate in the chamber.
Free electrons in the gap are accelerated by the electric field.
As they collide with air molecules, they create additional ions
and newly-freed electrons. The exponentially increasing elec-
trons and ions cause regions of the air in the gap to become
electrically conductive in a process called dielectric break-
down and finally an electrical discharge or an electric spark
occurs.
The ions or electrons produced during this process could be
captured on the falling oil droplets.
(1) Charge the droplets by pressing the piezo igniter.
The droplets are charged with unknown charge. Some could
have many electrons (or positive ions), some a few, and
some could have no charge. If you find too few droplets have
net charges, press the piezo igniter again.
Note
The object is to get a small number of drops, not a
large, bright cloud from which a single drop can be cho-
sen.
① To make oil droplets, squeeze the atomizer bulb with
one quick squeeze.
② Then squeeze it slowly to force the droplets through
the hole in the droplet hole cover, through the droplet
hole in the top capacitor plate, and into the space be-
tween the two capacitor plates.
Excessive use of the atomizer can cause too many
drops to be forced into the viewing area and prevent ob-
servation of drops. Besides, repeated squirts of the atom-
izer can cause the plate hole to be clogged and fail to
produce any drops in the viewing area. In such cases,
turn off the power supply, disassemble the chamber, and
then clean them with a soft tissue.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Millikan Oil Drop Experiment Ver.20170911
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 10 / 19
Step 4. Selection of the Drop
(1) Select an appropriate droplet for your measurement.
The following charts shows the graphs of Eq. (8) computed
with conditions of 𝑉𝑉 = 300 V, 𝑑𝑑 = 0.0074 m, 𝜌𝜌 = 886 kg m3⁄ ,
𝜋𝜋 = 1.862 × 10−5 Pa ∙ s (𝑇𝑇 = 25℃), and fall/rise dist.= 0.5 mm..
(Notice that graphs could be altered when your experimental
conditions vary.)
For example, if you select a droplet which requires 15 sec-
onds to fall 0.5 mm when the plates are not charged, and if
the droplet has 2, 3, 4, or 5 excess electrons, the droplet will
rise the same distance in the 15.7, 7.7, 5.1, or 3.6 seconds
when the plates are charged. If it has only 1 excess electron,
it will never rise. The droplets which have more than 6 excess
electrons are almost indistinguishable since there are small
differences in rise time.
A droplet which requires 5 seconds to fall 0.5 mm when the
plates are not charged, is good sample for measuring the
charge of 6𝑒𝑒, 7𝑒𝑒, 8𝑒𝑒, 9𝑒𝑒, or 10𝑒𝑒. It will take about 38.4, 15.7,
9.9, 7.2, or 5.7 seconds for the droplet to rise 0.5 mm again.
Thus, it is recommended to select the droplet for which it
takes 5-20 seconds to freely fall 0.5mm (plates not charged),
and for which it takes more than 5 seconds to rises 0.5mm
(plates charged).
If you have enough time for observations, you had better set
the moving distance at 1.0mm because you can decrease
measurement errors. However, you should also notice that a
long-time observation is not always good, since your droplet
could disappear from view during observation, or uncontrolla-
ble factors could disturb your observation.
If too many droplets are in view, charge the capacitor plates
and wait until many of them get out of sight. If you find that
too few droplets have net charges to permit the selection of
an appropriately sized and charged drop, press the piezo
igniter again.
When you find an appropriately sized and charged oil drop-
let, fine tune the focus of the viewing scope. (The oil droplet
is in best focus when it appears as a pinpoint of bright light.)
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Millikan Oil Drop Experiment Ver.20170911
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 11 / 19
Step 5. Collecting Data on the Rise and Fall of the Droplet
Measure the rise (plates charged) and fall (plates not
charged) velocities of the selected droplet about 5-10 times.
The greatest accuracy of measurement is achieved if you
time from the instant that the droplet passes the reticle line A
to the instant that the droplet passes the reticle line B (or C).
The distance between the reticle lines is 0.1 mm, so A and B
are 0.5 mm apart, and A and C are 1.0mm apart.
(1) With the plates not charged, start the stopwatch at the
moment the falling drop passes the line A.
(2) At the moment that the droplet reaches the line B or C,
record the lap time by clicking the [Lap] button of the stop-
watch, and apply electric fields at the same time.
(3) At the moment that the rising drop reaches the line A, rec-
ord the lap time again and stop applying electric fields at the
same time.
(4) Repeat step (2)~(3) 5~10 times.
distance(m) Fall time (s) Rise time (s)
1
2
3
4
5
…
average
(5) Calculate the average fall and rise velocities of the select-
ed droplet.
𝑣𝑣𝑓𝑓 = _______________ (m/s)
𝑣𝑣𝑟𝑟 = _______________ (m/s)
(6) Calculate the charge on the droplet.
𝑞𝑞 = 18𝜋𝜋𝑑𝑑𝑉𝑉�𝜋𝜋3𝑣𝑣𝑓𝑓3
2𝜌𝜌𝘨𝘨 �1 +𝑣𝑣𝑟𝑟𝑣𝑣𝑓𝑓� (8)
𝑞𝑞 : Charge carried by the droplet (C)
𝑉𝑉 : Potential difference across the plates (V)
𝑑𝑑 : Separation of the plates of the capacitor (m)
𝜌𝜌 : Density of oil (886 kg m3⁄ )
𝘨𝘨 : Acceleration of gravity (m s2⁄ )
𝜋𝜋 : Viscosity of air (Pa ∙ s = N ∙ s m2⁄ ) (See appendix.)
𝑣𝑣𝑓𝑓 : Velocity of fall without E-field (m s⁄ )
𝑣𝑣𝑟𝑟 : Velocity of rise without E-field (m s⁄ )
If the result of this first determination for the charge on the
drop is too great, you should use slower moving droplets in
subsequent determinations.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Millikan Oil Drop Experiment Ver.20170911
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 12 / 19
Step 6. Changing the Charge of the Droplet
It is desirable to observe as many different charges on a
single drop as possible. If the droplet is still in view, attempt to
change the charge on the droplet and measure the new rising
velocity many times, if possible.
(1) Bring the droplet to the bottom of the field of view using
the plate voltage switch.
(2) Rotate the plate charging switch to the charging position
to the droplet to rise.
(3) Change the charge on the droplet by pressing the piezo
igniter as described previously.
If the rising velocity of the droplet changes, make as many
measurements of the new rising velocity as you can.
(4) Repeat Step 5.
(5) If the droplet is still in view, repeat Step 6 as many times
as you can.
Step 7. Additional Measurement
Repeat step 1 to 6 with other droplets.
Repeat the experiment with at least 20 different charge
through the steps 5~7. It is desirable to get as many data as
you can.
Step 8. Analysis
Consider appropriate analysis methods to find the elemen-
tary charge with your experimental data. For example, the
following statistical approach can be considered when lots of
data collected.
The following chart shows an example of the measured
charges of 200 oil drops. Red dots are measured values and
blue lines represent theoretical values of 𝑛𝑛𝑒𝑒. Because of ran-
dom errors, your measurements may be deviated from an
expected value in every trial. Usually random errors follow a
normal distribution, so the assumption of normality can be a
good first approximation.
A histogram is a good tool giving a rough sense of the densi-
ty of the underlying distribution of the data. In the process of
dividing the entire range of charge values into series of inter-
vals, and then counting how many values fall into each inter-
val, we can get the following histogram. As the histogram
shows, the example data have a distribution that is nearly
normal at several peaks. Curve fitting can be used to find the
best fit curve, and finally the elementary charge is obtained.
See appendix for more instructions.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Millikan Oil Drop Experiment Ver.20170911
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 13 / 19
1. Viscosity
Sutherland’s formula can be used to derive the viscosity 𝜋𝜋
of an ideal gas as a function of a temperature
𝜋𝜋 = 𝜋𝜋0𝑇𝑇0 + 𝐶𝐶𝑇𝑇 + 𝐶𝐶 �
𝑇𝑇𝑇𝑇0�32
𝐶𝐶 : Surtherland’s constant
𝑇𝑇 : input temperature (K)
𝑇𝑇0 : reference temperature (K)
𝜋𝜋0 : reference viscosity (Pa ∙ s) at reference temperature 𝑇𝑇0
Surtherland’s constant and reference values for dry air are
𝐶𝐶 = 120
𝑇𝑇0 = 291.15 K
𝜋𝜋0 = 1.827 × 10−5 Pa ∙ s
2. Thermistor
A thermistor is a specialized resistor, intentionally designed
to be thermally sensitive and its primary characteristic is its
ability to alter its electrical resistance in response to changes
in case temperature. The resistance of a 10 kΩ thermistor at
𝑇𝑇(℃) follows the table below.
𝑇𝑇
(℃) 𝑅𝑅
(kΩ) 𝑇𝑇
(℃) 𝑅𝑅
(kΩ) 𝑇𝑇
(℃) 𝑅𝑅
(kΩ)
0 27.616 26 9.6306 36 6.6859
5 22.266 27 9.2768 37 6.4535
10 18.066 28 8.9879 38 6.2304
15 14.748 29 8.6132 39 6.0162
20 12.110 30 8.3020 40 5.8104
21 11.650 31 8.0037 45 4.8965
22 11.210 32 7.7177 50 4.1454
23 10.789 33 7.4435 60 3.0106
24 10.386 34 7.1805 80 1.6669
25 10.000 35 6.9281 100 0.97771
3. Analysis
The following table is an example of 200 observations of the
droplet charge. We now consider a statistical approach to find
the elementary charge using the data.
No. 𝑞𝑞 (× 10−19)
No. 𝑞𝑞 (× 10−19)
No. 𝑞𝑞 (× 10−19)
No. 𝑞𝑞 (× 10−19)
No. 𝑞𝑞 (× 10−19)
1 3.531 41 9.307 81 9.944 121 6.392 161 3.440
2 5.073 42 1.531 82 9.147 122 3.063 162 9.108
3 10.682 43 4.265 83 6.113 123 11.549 163 9.667
4 3.196 44 1.894 84 11.253 124 3.262 164 9.392
5 9.814 45 6.676 85 6.434 125 3.304 165 9.706
6 7.564 46 6.576 86 9.316 126 4.485 166 1.310
7 1.524 47 9.486 87 11.345 127 11.141 167 5.176
8 9.415 48 1.804 88 5.300 128 11.163 168 4.914
9 6.698 49 3.650 89 7.686 129 5.983 169 6.085
10 8.377 50 11.126 90 6.215 130 5.154 170 9.567
11 3.335 51 1.342 91 3.139 131 2.863 171 10.952
12 4.818 52 6.491 92 9.944 132 4.978 172 4.399
13 7.682 53 6.288 93 3.752 133 5.289 173 3.050
14 9.362 54 3.491 94 9.834 134 11.228 174 4.452
15 1.399 55 6.319 95 1.552 135 8.476 175 1.499
16 6.674 56 10.890 96 9.198 136 6.276 176 10.055
17 1.590 57 7.607 97 9.546 137 3.297 177 7.870
18 3.430 58 1.081 98 4.727 138 1.283 178 8.071
19 8.024 59 11.260 99 9.722 139 3.536 179 4.521
20 10.045 60 3.313 100 8.142 140 3.181 180 2.777
21 6.356 61 5.106 101 7.990 141 9.900 181 10.925
22 1.010 62 9.749 102 8.502 142 9.149 182 3.292
23 1.670 63 6.577 103 9.369 143 3.569 183 4.326
24 1.254 64 3.636 104 6.730 144 4.993 184 10.047
25 10.089 65 8.569 105 6.215 145 10.891 185 1.994
26 11.158 66 11.731 106 9.928 146 7.661 186 11.221
27 11.096 67 8.134 107 9.782 147 6.123 187 6.480
28 9.145 68 4.332 108 4.996 148 11.630 188 8.093
29 10.808 69 7.595 109 2.122 149 2.663 189 4.788
30 9.805 70 3.072 110 1.805 150 9.834 190 9.311
31 6.001 71 1.894 111 3.100 151 4.772 191 6.287
32 7.967 72 6.204 112 1.704 152 9.502 192 2.482
33 7.777 73 8.201 113 7.964 153 5.007 193 9.542
34 6.508 74 11.333 114 2.666 154 8.070 194 8.394
35 11.220 75 6.911 115 3.175 155 5.922 195 9.636
36 9.850 76 5.846 116 1.523 156 10.345 196 1.837
37 4.726 77 9.901 117 4.894 157 1.676 197 10.184
38 6.859 78 7.836 118 4.955 158 11.065 198 5.772
39 5.254 79 9.453 119 8.189 159 10.612 199 9.505
40 6.237 80 11.339 120 1.002 160 11.125 200 9.567
Appendix
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Millikan Oil Drop Experiment Ver.20170911
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 14 / 19
The following chart shows the distribution of the measured
charges of 200 oil drops as in the previous table. X-axis is
sequential number of droplets and y-axis is charge. Red dots
are measured charges of each oil droplet. Theoretical values
of 𝑛𝑛𝑒𝑒 are plotted in blue lines.
To create a histogram, we need a frequency distribution
table which contains the counts of the occurrences of values
within particular intervals or bins. The theory of statistics in-
troduces several rules for how to choose the optimal bin size.
However, in this example, we simply set it to 0.1602(× 10−19)
which is 1/10 of the elementary charge. Divide the range into
as follows and count how many values fall into each bin. Now
we have the following distribution table and histogram.
Bin Frequency
0 0
0.1602 0
0.3204 0
…
1.4418 4
1.602 6
1.7622 3
…
You can easily carry out the previous process using any
spreadsheet application such as ‘Excel’, which providing au-
tomated function to generate frequency table and histograms.
To create a histogram in Excel (the following guide applies
to Excel 2013 or prior versions), you need to load [Analysis
ToolPak] (분석 도구). Click [File] (파일) tab, click [Options]
(옵션), and then click [Add-Ins] (추가 기능) category. In the
[Manage] (관리) box, select [Excel Add-ins] (Excel 추가 기능)
and then click [Go] (이동).
In the [Add-Ins] (추가 기능) box, check the [Analysis Tool-
Pak] (분석 도구) checkbox, and then click [OK] (확인). Now
you can use [Data Analysis] (데이터 분석) in the [Analysis]
(분석) group on the [Data] (데이터) tab.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Millikan Oil Drop Experiment Ver.20170911
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 15 / 19
In one column, type bin numbers (evenly distributed inter-
vals) in ascending order. These represent the intervals by
which you want to measure the frequency.
Click [Data] (데이터) tab, click [Data Analysis] (데이터 분석)
and then select [Histogram] (히스토그램).
In the [Input Range] (입력 범위) box, enter the cell refer-
ence for the data range that has the droplet charges. In the
[Bin Range] (계급 구간) box, enter the cell reference for the
range that has the bin numbers. Under [Output Options] (출력
옵션), choose an output location. Check [Chart Output] (차트
출력) to shows an embedded histogram chart.
A frequency table and a histogram will be automatically
generated as follows.
The next step requires data analysis applications such as
‘Origin’ or ‘MATLAB’, since Excel does not provide complex
curve fit models such as Gaussian curve or multi-peak fitting.
When any data analysis app is not available, we recommend
the lightweight application, ‘MagicPlot’. (You can download
the free version of it at magicplot.com.)
(Continued)
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Millikan Oil Drop Experiment Ver.20170911
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 16 / 19
Run MagicPlot application and click [Create new Table] icon.
Type bin numbers in A column and frequency data in B col-
umn. (Or, use keyboard shortcuts such as Ctrl+C and Ctrl-V
to copy and paste data from Excel to MagicPlot.)
To create a Fit Plot, select your B column in Table, then click
[Create Fit Plot] button in the toolbar. Select [Line & Marker].
Now the following plot is generated.
To customize your plot, click [Line & markers] tap of [Curve
properties] window. Select [Vert Bars] for line type and check
[Fill] the lines.
To add a Fit Curve, click [Add] button in the [Fit Curves] tab
and select [Gaussian].
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Millikan Oil Drop Experiment Ver.20170911
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 17 / 19
Fitting process assumes that certain initial values of pa-
rameters are set before fitting. So you have to approximately
locate peaks before fitting. Using two circles on the fit curve,
drag the fit curve on corresponding position in data plot.
Add additional Gaussian curves and rearrange them. In the
example, the data plot has 7 peaks, so we added 7 fit curves.
Click [Fit by Sum] button. Then MagicPlot will fit the data
with the sum of each fit curves.
You can find the parameters of each Gaussian curves in the
[Fit Curves] tab.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Millikan Oil Drop Experiment Ver.20170911
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 18 / 19
Your TA will inform you of the guidelines for writing the laboratory report during the lecture.
Please put your equipment in order as shown below.
□ Delete your data files from your lab computer.
□ Turn off the lab computer.
□ With the voltage adjustment knob set at zero, turn off the power supply and unplug the power cable.
□ Unplug the dc adapter of the LED lamp.
□ Assemble the chamber. (Be careful not to lose any parts of it.)
□ Screw the focusing wire to its storage location on the platform.
□ Place the atomizer into the holder on the platform. (Handle the atomizer with care. It is very fragile.)
□ Do not unplug the high voltage patch cords.
Result & Discussion
End of Lab Checklist