Lab I: Coulomb Balance
George Wong
Julia Bladykas
Instructor: Geoffrey Ryan
Experiment Date: 7 February 2012
Due Date: 14 February 2012
1
Objective
The objective of this laboratory was to experimentally find the value of ε0 (the vacuum
permittivity constant) by measuring other variable values and evaluating them in a formula.
2
Theory
q1 q2
The force between two charges q1 and q2 in a vacuum is given to be F = 4π
2 , which
0R
ends up allowing for the generation of the formula for force between two charged plates (as
AV 2
where F is in newtons, A is area of the plates
in a capacitor) can be given by F = 02d
2
in meters2 , V is the voltage potential between the plates in volts, and d is the separation
between the plates in meters.
Because we wish to find the value of 0 , we rearrange the above formula and determine
2F d2
,
0 =
AV 2
which indicates that we will be able to find the value of 0 if we can determine the voltage
required between two plates of known area and known separation to induce a certain force.
From other studies of physics, we know that F = ma or, in the case of gravity, F = mg
wherein g ≈ 9.8 sm2 .
For the later analysis of error, we know that the sample standard deviation of a set of data
can be equated to the uncertainty in the measurement of the data. It is given by:
σ=
v
u
u
t
N
1 X
(xi − x̄)2 .
N − 1 i=1
An interesting note: a product of Maxwell’s equations suggests that the speed of light is
related to 0 and µ0 (vacuum permeability) by the formula c = √10 µ0 .
arearectangle = l · w
Given Fx,y,... , uncertainty in F is given by δF =
or δF = F ×
q
( ∂F
)2 + ( ∂F
)2 + · · ·
∂x
∂y
q
( δx
)2 + ( δy
)2 + · · ·
x
y
2
0 = 8.85 × 10−12 NCm2
3
Set Up
The primary apparatus was a coulomb balance—a device made of two plates of metal that
can be charged, vertically separated from each other, the bottom fixed, the top able to freely
1
move but counterweighted by a movable mass. The plates are charged by being connected
to a variable power supply running with a resistance box.
Also used over the course of the experiment were: laser, marking paper, plastic plate, small
precise masses, measuring equipment.
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Procedure
1. Balance Position Calibration
• The balance was set up and the counterweight was moved and set to a neutral
position.
• A plastic square was inserted between the two plates, the top plate was made to
rest down upon the square and bottom plate.
• The plates were readjusted/positioned so that they were level with each other and
the square.
• A laser was set up to point at a mirror attached to the top plate of the balance.
It was repositioned until it reflected back onto a sheet of paper attached to the
laser body.
• The location of the laser beam on the paper was marked.
• The height (or thickness) of the plastic square was measured and noted on Table
1 below.
2. Voltage-Force Matching
• A weight with known mass was placed on the center of the top plate, with all
other equipment off.
• The counterweight was moved until the laser marked the same position as before
(without mass).
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• With the mass removed, the voltage on the machine (and between the two plates)
was increased until the laser indicated that the separation between the plates was
the same as before with the mass. This voltage was recorded as on Table 2 below.
• The previous step was repeated several times for more data.
• This process was repeated for weights of different known masses.
3. Materials Replaced; Station Tidied
4. Data Analyzed
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Data
Table 1: Plate Dimensions
(m)
5x5 plate
width
.1272
length
.1272
thickness
.0020
error
± .0003
Table 2: Voltage for Mass Measurements (V)
mass (mg)
30
40
50
60
100
6
Trial 1
192
208
261
291
456
Trial 2
195
207
259
284
427
Trial 3
192
211
263
289
423
Trial 4 error
193
±2
209
±2
270
±2
286
±2
412
±2
Calculations
To calculate the area (and uncertainty) of the metal plates, we used the formula:
Arectangle = a = l · width
2
.1272m
q × .1272m = .01618m
2
2
δa = (δl · w) + (δw · l)
.00005m2 =
q
2 · (.1272 · .0003)2 m2 a = 1.618 ± .005 × 10−2 m2
From studies of mechanics, we know F = mg; therefore, we can write:
4.9 × 10−4 N = 5 × 10−5 kg · 9.8m · s2
···
Average is defined to be N
i xi /N
261 V + 259 V + 263 V + 270 V / 4 = 263 V
···
P
3
To find the
q error in 0 , we can use the following formula:
δ0 = 0 · ( δF
)2 + ( δd
)2 + ( δa
)2 + ( δV
)2
F
d
a
V
Table 3: Voltage and Force → 0
mass (mg) Force (N)
30
2.9 × 10−4
40
3.9 × 10−4
50
4.9 × 10−4
60
5.9 × 10−4
100
9.8 × 10−3
Avg Voltage (V)
193±2
209±2
263±2
288±2
430±2
d (m)
.0020 ± .0001
.0020 ± .0001
.0020 ± .0001
.0020 ± .0001
.0020 ± .0001
area (m2 )
(1.618±.005) × 10−2
(1.618±.005) × 10−2
(1.618±.005) × 10−2
(1.618±.005) × 10−2
(1.618±.005) × 10−2
average
0 ( N C·m2 )
3.9×10−12
4.4×10−12
3.5×10−12
3.5×10−12
2.6×10−12
3.6×10−12
Table 4: σvoltage and Definitions of σ0
mass (mg)
30
40
50
60
100
σvoltage (V)
1.4
1.7
4.7
3.1
19
σ0 ( N C·m2 ) — by σvoltage
3.9×10−13
4.5×10−13
3.7×10−13
3.6×10−13
2.6×10−13
4
σ0 ( N C·m2 ) — by σvoltage = 2
3.9×10−13
4.4×10−13
3.5×10−13
3.5×10−13
2.6×10−13
The plot on the top returned a value for 0 = 3.1×10−12
returned 0 = 2.3 × 10−12 N C·m2 .
C
N ·m2
whereas the plot on the bottom
The following section shows the python code used to calculate the values for 0 and its
uncertainty. No written calculations were done; therefore, none are otherwise shown.
# 14 feb 2012 : LAB I
import numpy as np
import matplotlib.pyplot as plt
def ChiSqFit(x,y,u):
’’’ Returns slope and y-int of linear fit to (x,y,u) data set ’’’
xhat = (((x/(u**2))).sum())/(((1/(u**2))).sum())
yhat = (((y/(u**2))).sum())/(((1/(u**2))).sum())
slope = (((x-xhat)*(y/(u**2))).sum())/(((x-xhat)*(x/(u**2))).sum())
yint = yhat - slope*xhat
s_b = np.sqrt(1/(((x-xhat)*(x/(u**2))).sum()))
s_a = np.sqrt((s_b**2)*((((x**2)/(u**2)).sum())/(1/(u**2)).sum()))
return slope, yint, s_a, s_b # returns m, b; sig_a, sig_b
def eps(mass,volt,volt_unc):
dist = .002
dist_unc = .0001
area = .01618
area_unc = .00005
force = mass*9.8
force_unc = 0
top = (2*force*dist*dist)
top_unc = np.sqrt((4*force*dist*dist_unc)**2+(2*dist*dist*force_unc)**2)
bot_unc = np.sqrt((2*area*volt*volt_unc)**2+(volt*volt*area_unc)**2)
bot = (area*volt*volt)
eps_0 = top/bot
#eps_0_unc = np.sqrt((2*dist*dist/area/volt/volt*force_unc)**2+
(4*force*dist/area/volt/volt*dist_unc)**2+(2*force*dist*dist/
5
volt/volt*area_unc)**2+(4*force*dist*dist/area*volt*volt_unc)**2)
eps_0_unc = eps_0 * np.sqrt((force_unc/force)**2+(2*dist_unc/dist)**2
+(area_unc/area)**2+(2*volt_unc/volt)**2)
return eps_0,eps_0_unc,top,bot,top_unc,bot_unc
#input section
’’’
mass_mg = input(’Enter mass in mg: ’)
mass = mass_mg*10**(-6)
mass = np.array(mass)
volt = input(’Enter voltage in v: ’)
volt = np.array(volt)
volt_unc = input(’Enter volt_unc in v: ’)
#volt_unc = 2 #(we de-comment this if we’re forcing said uncertainty)
volt_unc = np.array(volt_unc)
’’’
# ORIGINAL DATA
mass_mg = [30,30,30,30,40,40,40,40,50,50,50,50,60,60,60,60,100,100,100,100]
mass_mg = np.array(mass_mg)
mass = mass_mg*10**(-7)
volt = [192,195,192,193,208,207,211,209,261,259,263,270,291,284,289,286,456,
427,423,412]
volt = np.array(volt)
volt_unc = [1.4,1.4,1.4,1.4,1.7,1.7,1.7,1.7,4.7,4.7,4.7,4.7,3.1,3.1,3.1,3.1,
19,19,19,19]
volt_unc = np.array(volt_unc)
#volt_unc = 2
# do the stuff
eps_0,eps_0_unc,ys,xs,ys_unc,xs_unc = eps(mass,volt,volt_unc)
m, b, sa, sb = ChiSqFit(xs,ys,xs_unc) # we dont’ care about sa and sb
fitline = m*xs+b
plt.plot(xs,ys,’ok’,markersize=2)
plt.errorbar(xs,ys,yerr=ys_unc,xerr=xs_unc,ecolor=’grey’,fmt=None)
plt.plot(xs,fitline,’:c’)
plt.ylabel(’ $2Fd^2$ ’)
plt.xlabel(’ $AV^2$ ’)
plt.title(’Finding \epsilon_0$ with uncertainty = 2’)
plt.show()
print ’fit slope: ’, m
print ’eps_0: ’, eps_0
print ’eps_0_unc: ’, eps_0_unc
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7
Error Analysis
We took the error in the reported masses of the weights to be zero, which led to an uncertainty
in force of zero (force = mass × g). This, obviously, was not exactly the case; however, its
effect was assumedly negligible.
At one point, we took the sample standard deviation of the set of data for voltage, later
labeling it as the uncertainty in the voltage. Theoretically, this value could have been used
as uncertainty for later calculations; however, I decided not to use it, because I believed
that it would allow for misleading conclusions. The basis for my reasoning is that there was
random error in many different ‘environmental variables’ that changed from one trial to the
next. The one part of the equation that did not change from one trial to the next was,
ironically, 0 . From this point of view, we can see that we were actually measuring ratios
of 0 . Seeing as we wished to derive 0 from the values we were measuring, it would have
been foolish to use the standard deviation of the voltage data, because that would, in part,
account for the error in the other valued measures—it would not only compensate for the
error in voltage, but also in distance, force, etc. (we remember that these can nearly be
defined in terms of 0 ).
Needless to say, this necessitated assuming the uncertainty/error in the voltage was something
else, for which I chose the value 2 V, because it allows for some error above what the machine
should give (only ± 1 V) while not suggesting wild inaccuracy. For reasons explained later
in the conclusion, the process was repeated for δ0 defined using the standard deviation.
In the end, it was found that the average value we found for 0 was off by around 60% from
the accepted value; however, this result might be a bit misleading. Although it was off by
60%, it was off by under one order of magnitude. When measuring values in these sorts of
ranges, that might be considered satisfactory or not. For this lab, such error was considered
to be satisfactory; it was also specifically said to be expected. Sadly, even when we consider
the uncertainty in our measured value of 0 , the overall error does not change significantly
(at best, we reduce the error to ≈ 44%.
Due to the nature of the experiment, error and uncertainty were able to manifest in a multitude of ways.
• THere was the assumption that the two plates were always perfectly aligned with each
other, presenting the area suggested for voltage difference.
• The voltmeter could only measure with a precision of ± 1 volt, further this imprecision
was overshadowed by the fact that there was a range of values that worked for several
trails.
• Working with such small masses meant that the tiniest speck of dust could majorly
offset the data.
• Because of the small materials and slight masses involved, air currents were easily able
to upset the data.
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• As the laser drew closer to the mark, it would move in a more jittery pattern, due to
the forces acting on it.
• The way in which the force was increased meant that there would need be much finer
control as the position grew closer to correct; however, there was no way to fine-tune
the system.
In the future, to increase the precision of the experiment and reduce error:
• Have the laser-marking paper be as far away as possible. This will increase visibility
in position change.
• Eliminate air currents (by use of vacuum box, etc.).
• Increase the resolution of the voltmeter/step by which it changes.
• Increase the size of the plastic plate used to separate the two metal plates at the
beginning of the experiment.
8
Question
“In placing the 50 mg mass on the top plate, you are advised to place this mass in the center
of the top plate. Would you get the same result if the mass were not placed in the center,
but perhaps closer or further away from the balance beam? Discuss.”
The results would be different, because the force being applied to the top plate would ultimately be different (if not by much). The nature of the pivot meant that placing the mass
at a point not-center would cause one part of the plate to move different than the other.
As such, when calibrating with the laser position system, the position would not necessarily
be representative of the same separation distance found with the plate of known thickness.
Much of the experiment depends upon this distance measurement; followingly, the difference
would most likely greatly offset the data.
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Conclusion
The purpose of this lab examination was to experimentally determine the value of 0 by using
a coulomb balance and a voltmeter/power supply to measure the amount of voltage necessary
to deliver a specific force (predetermined by counterweighting procedures).
Ultimately, the purpose of this lab was fulfilled: we were able to experimentally calculate
a value for 0 by making measurements of other quantities. Because we had a known or
2
accepted value of 0 (≈ 8.85×10−12 NCm2 ), we were able to determine how accurately we were
performing the experiment. By calculation, we determined that there was just under 60%
error in our measurements and that the value was off by under one order of magnitude.
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Also, looking at the graphs of ‘top of equation’ versus ‘bottom of equation’, we can see that a
linear relationship is present. This is good, and is what we want. We note by looking at the
two graphs and the values for 0 produced by them that the one using standard deviation as
uncertainty in voltage produced a much better analogue of what was being observed. This
makes sense; however, it may not necessarily be completely representative of the data set
that was actually taken. While it better fits with our preconceived notions about 0 , this is
probably not ‘coincidental’.
Due to the intensely delicate nature of many of the operations performed over the course
of the experiment, the data taken were not altogether bad at all. The inaccuracies were to
be expected, and the fact of the matter is that the important concepts regarding 0 were
revealed: its linearity under certain circumstances; its constancy. . .
In future experiments of the same nature, it might be a good idea to suggest which masses
to continue on with after having performed the experiment with 50mg. Further, exact specifications regarding how many of each type of trial should be made would aid in the accuracy
of the results. It might also be interesting to study the relationships involved in the formula
used for the derivation of 0 . That is, how does the force drop off inversely proportional to
distance, how does varying the force affect the graphs produced; the voltage; etc..
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