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Use of a parallel plate capacitor to measure the numerical value of the permittivity of free space ε𝑜 .
Shawn Mayer
Jake Wojokowski, Demi Xenos
Department of Physics, Canisius College
Abstract:
The permittivity of free space provides a relationship between the capacitance of a parallel plate capacitor, the area of the
system, and the height of the separation between the two boards. Different sized fiber boards covered in aluminum foil were
used as capacitors. A capacitance meter was used to measure capacitance of the system. Multiple boards and separators were
used to test this relationship. We found reasonable agreement between our result which was 8.59x10-12 and the desired
constant which is 8.85x10-12.
Introduction and Theory:
The electric field of a uniformly charged sheet is constant at any distance from the sheet. When two
plates of different charge are placed near each other, the two electric fields between the plates add
while the electric field outside the plate cancel. When the plates are close to each other to form a
capacitor, the electric field between the plates is constant throughout the interior of the capacitor as
long as one is not near the edge of the plate [2]. The electric field that is formed is a result of the charges
across the two plates separated by a certain distance.
𝐸=
∆𝑉+
,
𝑑
(1)
Where E is the electric field, delta V is the change in voltage and d is the distance.
∆𝑉+ = 𝑉+ − 𝑉− ,
(2)
Capacitance measures the relative amount of charge that can be stored on a pair of parallel plates for a
given amount of voltage.
𝐶=
𝑄+
,
∆𝑉
(3)
Where C is the capacitance, Q is the charge on the positive plate and delta V is the change in voltage. If
the capacitance increases, then more charge can be stored when the same potential is applied [1]. To
determine the capacitance of an object, a relationship was created with respect to the objects structure
giving a method to determine the amount of charge a capacitor could hold. To confirm this relationship
we constructed a parallel plate capacitor as shown in figure 1 and determined what the value of the
constant was that related capacitance to the dimensions of the parallel plate capacitor.
We were able to take the measurements from our system
𝐴 = 𝐿(𝑊), (4)
where L is the length of the plate, W is the width of the plate and A is the area of the plate. Then using
the result from equation 4 we were able to take another measurement from our system along with a
reading from the capacitance meter.
𝐴
𝐶 = ε𝑜 , (5)
𝑑
Figure 1. The experimental setup of the parallel plate capacitor
where C is capacitance, d is the height of the separators in between the two plates and ε𝑜 is the
permittivity of free space to directly determine the value of the permittivity of free space. We predicted
that with the measurements taken, using a capacitance vs area/distance graph to analyze the data
would result in a linear slope that would equal the value for the permittivity of free space.
Procedure:
As a group we decided that we would set our sample size at three different sized plates; 0.09m², 0.04m²,
and 0.0225m² and three different separator heights; 0.002m, 0.010m, and 0.0062m. We used fiber
boards covered with aluminum foil as the parallel plate capacitor and broken popsicle sticks as the
separators. For each trial we began by measuring the length and width of the fiber board with a ruler to
find the area of the board. Then we measured the height of the separators in between the two boards
using calipers. We had to make sure that the value of the height of the separators was much less than
the length of the board. Because if this wasn’t the case, the electric field bulging at the end of the plate
would have increased resulting in an increased C value. We then attached the wires from the
capacitance meter to the piece of aluminum foil and recorded the readings displayed on the capacitance
meter. All measured quantities are shown in Table 1 below.
Capacitance (F)
1.29x10-10
7.78x10-11
3.70x10-10
1.84x10-10
3.53x10-11
2.88x10-11
1.01x10-10
2.83x10-11
1.28x10-11
2.06x10-10
Area (m²)
0.09
0.09
0.09
0.04
0.04
0.04
0.0225
0.0225
0.0225
0.09
Distance(m)
0.0062
0.010
0.002
0.002
0.0062
0.010
0.002
0.0062
0.010
0.020
Table 1 Recorded measurements from each trial performed
Finally, we used our recorded information and made a graph of the capacitance vs the area/distance on
Microsoft Excel.
Analysis and Results:
Using equation 5 and the values we had recorded we knew that if we made a graph with capacitance on
the y-axis and area/distance on the x-axis that the slope would be a straight line which represented the
value of the permittivity of free space constant. Our graph is shown in the figure below.
Capacitance
4.5E-10
4E-10
y = 8.59E-12x - 2E-12
Capacitance (F)
3.5E-10
3E-10
2.5E-10
2E-10
1.5E-10
1E-10
5E-11
0
0
10
20
30
40
50
Area/Distance (m)
Figure 2. Results from all trials with the slop representing the permittivity of free space.
Our results show that a parallel plate capacitor with large plates and small distances between them will
result in large capacitance values and therefore make it be able to hold large amounts of charge. Also as
we predicted, the slope of the graph was a straight line and represents the value we were trying to
obtain. The value that we obtained for the permittivity of free space (8.59x10-12) is fairly close to the
accepted value which is 8.85x10-12 [3] (Percent error of 2.9%). The deviation between our result and the
accepted value is most likely due to errors in measurement using the calipers or ruler.
Conclusion:
In this experiment we have shown that the capacitance of a parallel plate capacitor is correlated with its
structure and what the value is that links them together. Our obtained value for the permittivity of free
space was slightly lower than the accepted value. With an increased number of trials, a more diverse
choice of plates and separators and more precise measuring equipment a more precise value could be
obtained experimentally.
Works Cited:
[1] Colwell, Catharine. PhysicsLAB. N.p., n.d. Web. 24 Mar. 2015.
<http://dev.physicslab.org/Document.aspx?doctype=3&filename=DCcircuits_CapacitorProperties.xml>.
[2] Lashway, Cherie. PhysicsNet. Western Washington University, the Physics Department, the Bureau of Faculty Research, n.d.
Web. 24 Mar. 2015. <http://faculty.wwu.edu/~vawter/physicsnet/topics/Capacitors/ParallCap.html>.
[3] "Permittivity." Encyclopædia Britannica. N.p., 12 Dec. 2014. Web. 25 Mar. 2015.
<http://www.britannica.com/EBchecked/topic/452314/permittivity>.
The abstract does meet Dr. Selkowitz’s criteria, it introduces what the paper is about i.e.
the two species and how one is considered a threat and the other isn’t and how to tell them apart.
It also explains why the study is important which is so that regulatory agencies can discriminate
between the two and take appropriate action when needed. It then briefly mentions the
conclusions of the experiment which was so that through close observation of different parts of
each species, it is possible to tell the difference between them.
The hypothesis of the paper was that analyzing surstylus and aculeus shape and size
using geometric morphometrics would be a successful method to increase success in
discriminating between the Rhagoletis pomonella and Rhagoletis zephyria.
The flies were collected from infested apples and snowberry fruit in Washington State.
The apples were collected from the ground and the snowberry from the plant. The fruit was
brought to Washington State University and from them the flies were harvested and then allowed
to mature. The flies then died off and were stored in 70% ethanol. The flies were separated into
groups according to the host fruit and sampling area and the data collected was organized
according to those groups.
The main statistical test used to address the hypothesis was the measurements of the
centroid size and length of the surstylus of the two species. The classification based on said
measurements were determined using CVA’s and assignment tests on the different groups.
Figure four shows the result clearly.
I believe that the hypothesis was proved as figure four clearly shows that using the
specified measurements, differentiating between the two species is simple. The two clusters
show that the measurements are independent of each other with little to no overlap.
The story in the apple maggot paper was about some flies that were eating all the apples
in an area where people did not want them to. So they set up paper traps that would end up in a
decrease in population of those flies. But the problem was that other flies that were not feeding
on the apples in question were also getting caught in said traps because of the close proximity to
their snowberry plants. So the people wanted a better way to differentiate between the two
species as to be able to place their traps in more advantageous locations to only target the pests.
To achieve this, the people acquired a sample size of each species and measured a part of their
anatomy which showed a clear and distinct difference in the two species. People would care
about this because the population of the flies that were infesting the apples would go down and
the people would be able to eat those apples instead of them being breeding grounds for insects.