Millikan Oil-Drop Experiment
Darrell Marquez
August 24, 2021
1
Objective
To measure and determine the elementary charge of an electron by balancing
three specific forces.
2
Idea
The overall idea of this experiment was to reproduce (as best we could)
R.A. Millikan’s Oil-Drop experiment, though, our setup differed greatly. Utilizing 21st century technology we were able to use a CCD video camera rather
than a microscope, and instead of oil, our solution was a mixture of micron-sized
latex spheres. Despite these changes our purpose remained the same.
The downward force of gravity, upward force of drag, and the varying direction of a uniform electric field were all used to manipulate our “captured”
latex spheres. These were the bedrock of our experiment yet other constants
also needed to be resolved, such as: the atmospheric pressure at our elevation,
the viscosity of air (our viscous medium), and separation of the parallel plates,
all of which were fixed. The only controls remaining were potential difference
(applied voltage) and the fall radius of the “captured” spheres. These were
composed to calculate a terminal velocity, which acted as the key ingredient in
determining our spheres’ radii and consequently their respective charges.
This experiment relies heavily on Stoke’s Law. Though, a correction to
Stoke’s Law must, however, be made.
Fd = −6πηrv = −Kv; K =
6πηr
pπηr
⇒ Fd = −
V
b
b
1 + pr
1 + pr
From the previous equations we can substitute them into one another and
solve for r, if we do this we obtain the following equation.
s

2 b
b 
1  18ηv
r=
+
−
2
gρ
p
p
This equation gives r based on a variable velocity(v). The charge(q) of an
electron is dependent upon which direction the electron is traveling. As for
this experiment we focused on the charge of the falling velocity(v), given by
1
q=
d
V
∗
6πηr
b
1+ pr
(Vr + Vf ). We can rearrange this equation to relate charge(q) to
a suspension voltage(Vs ), given by q =
3
Apparatus
4
Procedure
4πr 3 ρgd
3Vs .
1. Calibrate the scale on the television screen. Point the camera toward the
1mm calibration scale and adjust the distance between the camera and
target.
2. Focus the camera on a small wire and adjust the light source for maximum
illumination. Adjust the camera so that neither capacitor plate is visible
on the television screen and so that the sample chamber is centered in the
field of view.
3. Place the nozzle of the atomizer in the hole in the top plate and squeeze
gently. Then seal the chamber.
4. Watch the television screen to find a ball that is falling about 10mm in
60 seconds. Turn on the magnetic field with a positive current so that the
ball travels towards the top of the television screen to capture the ball.
5. Place your head level with the television screen to avoid parallax.
6. Adjust the voltage so that the ball is suspended and record the suspension
voltage.
7. Adjust the focus of the camera to keep the ball in focus.
8. Adjust the voltage so that the ball is traveling again.
9. Record the fall time of the ball with a negative current. Then switch to
positive current and record the rise time of the ball.
10. Record a total of 5 rise times and 5 fall times.
2
11. Either find a new ball or turn the ultraviolet light on for 10 seconds to
change the charge of the ball.
12. Take 10 different sets of 5 rise and fall times for new balls or balls with
changed charges.
5
Data Analysis and results
5.1
Pre-lab Questions
1. The following equation shows the relationship between suspension voltage
(Vs ) and the amount of charges on a sphere (n).
Vs (n) =
−4πρr3 gd
3ne
Graphically represented below:
2. The next equation shows the relationship between velocity and radius of
the sphere.
b
2ρgr2 (1 + pr
)
V =
9η
5.2
Data
With a radius(r) = 5.535 ∗ 10−7 m, and a velocity(v) = 4.89 ∗ 10−5 m
s
Fall velocity was calculated by dividing the distance by the time it took for
the sphere to fall that calculated distance as shown below.
Vf =
d
2mm
mm
= Vf =
= −0.38
tf
5.3s
s
While the radius was calculated using the following equation.
s

2 1
18ηv
b
b 
+
−
r= 
2
gρ
p
p
3
−5
torr*m, p = 314.725 torr,
We note that η = 1.828*10−5 Nmsec
2 , b = 6.17*10
kg
m
−3 m
g = 9.796 s2 , and ρ = 1050 m3 , and v = 2.94*10 s . Once we put it all together,
we calculate that r =1.64*10−6 m.
5.3
Sphere 1
The following table represents the first sphere we were able to isolate, we
had control of the potential and the distance that the sphere traveled.
Trial
Rise
Time [s]
±0.6
Fall
Time [s]
±0.6
Voltage
[V] ±1
Susp.
Voltage
[V] ±1
Distance
[mm]
±0.1
1
2
3
4
5
Average
30.37
23.81
26.68
25.09
32.75
27.7
5.84
5.06
5.25
5.06
5.34
5.3
192
192
192
192
192
192
134
134
134
134
134
134
2
2
2
2
2
2
5.4
Fall
Velocity
[ mm
s ]
±0.05
Radius
[µm]
±0.12
0.38
1.64
Sphere 2
The following table represents the second sphere we were able to isolate, we
had control of the potential and the distance that the sphere traveled.
Trial
Rise
Time [s]
±0.6
Fall
Time [s]
±0.6
Voltage
[V] ±1
Susp.
Voltage
[V] ±1
Distance
[mm]
±0.1
1
2
3
4
5
Average
14.50
14.65
15.18
14.90
14.93
14.8
0.59
0.68
0.62
0.62
0.72
0.6
119
119
119
119
119
119
105
105
105
105
105
105
1.9
1.9
1.9
1.9
1.9
1.9
5.5
Fall
Velocity
[ mm
s ]
±0.05
Radius
[µm]
±0.12
2.94
4.75
Sphere 3
The following table represents the third sphere we were able to isolate, we
had control of the potential and the distance that the sphere traveled.
4
Trial
Rise
Time [s]
±0.6
Fall
Time [s]
±0.6
Voltage
[V] ±1
Susp.
Voltage
[V] ±1
Distance
[mm]
±0.1
1
2
3
4
5
Average
10.50
10.68
10.53
10.31
10.42
10.5
0.59
0.68
0.62
0.62
0.72
0.6
131
131
131
131
131
131
118
118
118
118
118
118
1.9
1.9
1.9
1.9
1.9
1.9
5.6
Fall
Velocity
[ mm
s ]
±0.05
Radius
[µm]
±0.12
3.01
4.81
Sphere 4
The following table represents the fourth sphere we were able to isolate, we
had control of the potential and the distance that the sphere traveled.
Trial
Rise
Time [s]
±0.6
Fall
Time [s]
±0.6
Voltage
[V] ±1
Susp.
Voltage
[V] ±1
Distance
[mm]
±0.1
1
2
3
4
5
Average
9.53
9.50
9.18
9.21
10.28
9.5
0.53
0.75
0.68
0.60
0.56
0.6
131
131
131
131
131
131
109
109
109
109
109
109
1.9
1.9
1.9
1.9
1.9
1.9
5.7
Fall
Velocity
[ mm
s ]
±0.05
Radius
[µm]
±0.12
3.04
4.83
Sphere 5
The following table represents the fifth sphere we were able to isolate, we
had control of the potential and the distance that the sphere traveled.
Trial
Rise
Time [s]
±0.6
Fall
Time [s]
±0.6
Voltage
[V] ±1
Susp.
Voltage
[V] ±1
Distance
[mm]
±0.1
1
2
3
4
5
Average
18.56
19.18
18.75
18.62
17.96
18.6
0.59
0.56
0.65
0.62
0.53
0.6
127
127
127
127
127
127
123
123
123
123
123
123
1.9
1.9
1.9
1.9
1.9
1.9
5
Fall
Velocity
[ mm
s ]
±0.05
Radius
[µm]
±0.12
4.98
4.98
5.8
Spheres 6-10
The following table represents the same captured sphere, but with a changed
charge (via UV light) accompanied with each new measurement. We decided to
change the method once we were informed that our previous method was a tad
extraneous.
Trial
Rise
Time [s]
±0.6
Fall
Time [s]
±0.6
Voltage
[V] ±1
Susp.
Voltage
[V] ±1
Distance
[mm]
±0.1
1
2
3
4
5
Average
19.3
17.5
16.1
9.3
15.9
0.6
0.6
0.6
0.6
0.5
148
158
178
207
238
131
148
167
187
224
1.9
1.9
1.9
1.6
1.6
5.9
Radius
[µm]
±0.12
4.85
5.11
4.85
4.56
4.82
4.52
n-Values for Spheres
Sphere
Susp.
Voltage
[V] ±1
1
2
3
4
5
6
7
8
9
10
134
105
118
109
123
131
148
167
187
224
5.9.1
Fall
Velocity
[ mm
s ]
±0.05
3.06
3.39
3.06
2.74
30.1
Distance
Radius
between
[m]
±0.12*10−6 plates [m]
±10−7
−6
1.64*10
6.0198*10−3
−6
4.75*10
6.0198*10−3
−6
4.81*10
6.0198*10−3
4.83*10−6 6.0198*10−3
4.98*10−6 6.0198*10−3
4.85*10−6 6.0198*10−3
5.11*10−6 6.0198*10−3
4.85*10−6 6.0198*10−3
4.56*10−6 6.0198*10−3
4.82*10−6 6.0198*10−3
Sphere
Density
kg
[m
3]
g [ sm2 ]
Charge [C]
n
1050
1050
1050
1050
1050
1050
1050
1050
1050
1050
9.796
9.796
9.796
9.796
9.796
9.796
9.796
9.796
9.796
9.796
-8.54*10−18
-2.65*10−16
-2.45*10−16
-2.68*10−16
-2.60*10−16
-2.26*10−16
-2.34*10−16
-1.77*10−16
-1.32*10−16
-1.30*10−16
-54
-1652
-1527
-1673
-1625
-1410
-1459
-1106
-821
-809
Frequency of n
The following is a graph depicting the frequency of charges calculated.
6
5.9.2
Error
The n-values were calculated using the following equation, the data used
came from the above table.
−4πρr3 gd
n=
3Vs e
From the data we were able to calculate our errors, we found:
Statistical Error - uncertainty of the slope which is equal to:
3 ∗ 10−34 C
Systematic Error:
5 ∗ 10−19 C
The following graph represents our values of e − charges. The slope of the
best-fit-line is:
e = −1.60219 ∗ 10−1 9
7
6
Conclusion
A large majority of the error came from the spheres themselves. As they
were atomized, they would come out of the tube stuck together, which I believe
accounts for our extremely large n-values. The spheres may have also gained
additional charge from the tube they were propelled from. There were many
areas of potential error, such as having to keep in parallax with the television,
while also having to communicate to someone else when to start and stop a
timer. This presents a very large source of error: our reaction times. These
errors were accounted for by independently measuring the time taken for one to
press the timer according to another’s command. Then latter errors propagated
in great amount all throughout the data.
8