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Parallel-Plate Capacitor Measuring the Value of the Permittivity of Free Space (𝜀0 )
Amir (The Genghis) Khan
Syed (The Goose) Kamal
Abstract:
This experiment utilized three square parallel plate capacitors, which were placed in a closed circuit. The
capacitance (C) was measured by a capacitance meter. All three of the capacitor plates were different in length,
resulting in different areas. Popsicles sticks were used to vary the separation between the two plates. A value of 𝜀0 ,
was calculated through the knowledge of the relationship between 𝜀0 and our measured variables. The Slope
𝐴
obtained from the graph of C vs. was determined to be the calculated value of 𝜀0. The experimental value of 𝜀0
𝑑
was 1 x10^-11 was relatively close to the literature value of 8.85 x10^-12 F/M.
Introduction:
A parallel plate capacitor was constructed in this experiment, as shown in figure 1. Changing the
distance separating the plates of the capacitor, (area of the capacitor plate) determined the
association between capacitance, distance and area. The distance was manipulated and changed
by mini chips of the same size, which kept the area of capacitance constant. Parallel plate
capacitors produce a uniform electric field between the two plates. The permittivity of free space
(𝜀0 ) is used to relate units of electrical charge to mechanical quantities (area and length). To
calculate the numerical value 𝜀0, the capacitance had to be calculated, as well as the area and the
distance between the capacitance. This constant aided in calculating forces on particles that
would be impossible to obtain by mechanical measurements.
Figure 1. Homemade Parallel-Plate Capacitor
Theory:
The choice to use a parallel-plate capacitor was due to its ability to generate a uniform electric
field. Also parallel plate capacitors are devices in which the data is relatively easy to gather.
Ideally, a parallel-plate capacitor would be infinitely long, resulting in a perfectly uniform
electric field. Real capacitors are not infinite in size, but when certain conditions are met, a
homemade parallel-plate capacitor can act ideally.
Going back to Lab 2: Electric Field Mapping, it was observed that as the distance between
electrodes increased the electric field became weaker. As an electric field becomes weaker, it
becomes less uniform due to the separation of the electric field vectors. As distance between the
plates decreased, to a point where the separation distance was much smaller that the length of the
electrode, the bowing of the electric field became negligible.
That being said, Lab 2: Electric Field Mapping proved that there was a relationship between
electric field and distance from the source. This observation gave importance of 𝜀0 as a
physically relevant quantity, and allowed the measurement of the amount of field (twodimensional or three-dimensional) from a charged particle.
A crucial limitation for the experiment was that the length of the side of an electrode had to be
much larger than the distance separating the two plates. The ratio of distance-to-length must be
much less than one. This allowed us to assume that the electric field has planar symmetry, and to
ignore the bowing effect of the electric field.
College Physics defines capacitance as, “the constant of proportionality C between charge on the
capacitor and the potential difference between the two plates, the equation is given below
𝑄
C=𝛥𝑉 ,
(1)
𝐶
Where C is the capacitance (in farads), Q is the magnitude of charge (in Coulombs) on the plates,
and 𝛥𝑉𝐶 is the potential difference (in volts) between the electrodes. Capacitance is proportional
to the magnitude of charge on the plates and inversely proportional to the potential difference.
From previous experiments, it is known that the potential difference is determined by the
electrical field between the plates, due to the differences in charge on each electrode. The
distance and electric field are proportional to each other. If charges are packed farther away, the
field will be smaller, and vise versa. Therefore, we are able to define the electric field in a
parallel-plate capacitor with the following equation:
𝑄
Ecapacitor = 𝜀 𝐴, (from positive to negative) (2)
0
Where Ecapacitor is measured in Newtons per Coulomb (N/C), Q is the capacitors plate charge, A is
the area of the plate, and 𝜀0 is the permittivity constant. From Lab 2: Electric Field Mapping, we
observed that electric field strength of a uniform electric field is related to potential difference
and separation distance for equipotential surfaces providing the formula:
E=
𝛥𝑉𝐶
𝑑
,
(3)
The equation then becomes:
Q=
𝜀0 𝐴
𝑑
𝛥𝑉𝐶,
(4)
When equation 1 is compared to equation 4, we can determine the parallel plates capacitance as
C=
𝜀0 𝐴
𝑑
,
(5)
Where C is the capacitance (in Farads), 𝜀0 is the permittivity of free space constant (in F/m), A is
the area of the plates (in m2), and d is the separation between the plates (in m).
This equation allows us to infer the following relationships; capacitance is proportionate to the
area of the electrodes, and inversely proportionate to the distance separating the two plates.
Size of
Capacitor
plates
Large
Area of Capacitor
( 2
m)
Distance between
electrodes (m)
Area/Distance
Capacitance (F)
between electrodes
(m)
.090
2.9
31.0
4.6 x10^-10
4.3
20.9
1.5 x10^-10
6.5
13.9
9.5 x10^-11
Medium
.038
2.9
13.11
1.9 x10^-10
4.3
8.84
5 x10^-11
Small
2.9
7.75
1.2 x10^-10
.
022
4.3
5.23
3.9 x10^-11
6.5
3.46
3.0 x10^-11
Table 1: The Capacitance of parallel plate capacitors with various Areas and Separation
Since the spacing between the two plates did not affect the electric field, and the charges on the
plates were opposite but equal, it was assumed that the electric was uniform.
To solve for ε0
ε0 =
𝐶𝐴
𝑑
,
(6)
Procedure:
This experiment consisted of using three pairs of fiberboard squares covered in aluminum foil.
Each square differed in size (small, medium, and large) were used to set up a parallel plate
capacitor. Popsicle sticks were utilized in order to separate the plates. The popsicle sticks were
placed at each corner and in the middle of the apparatus. The average width of a popsicle stick
was calculated from measuring the thickness of three popsicle sticks, done by a caliper tool.
Initially the “small” board’s length was measured with a meter stick, and the area was calculated,
(process was repeated for the other two sized boards). Using the “small” boards and one popsicle
stick of separation, a parallel plate capacitor was assembled. A weight was placed on top of the
board to ensure that the surfaces of the electrodes were parallel. A capacitance meter was utilized
to measure capacitance; these values are recorded in Table 1. This method was repeated for both
the “medium” and “large” fiberboard squares covered in aluminum foil, with varying popsicle
sticks (1, 3, and 5).
The results were limited, due to the unsteadiness of the popsicle sticks to stack on one another.
Also the limited amount of capacitor could have limited the data.
𝐴/𝑑 vs. C
5E-10
4.5E-10
4E-10
y = 1E-11x - 4E-11
C (F)
3.5E-10
3E-10
2.5E-10
2E-10
1.5E-10
1E-10
5E-11
0
0
5
10
15
20
25
30
35
A/d (m)
Figure 2. Graph of relationship between capacitance and ratio of area-to-separation of an
electrode. (Linear)
Analysis:
Our predictions were demonstrated with the experimental results. It was displayed that a larger
area leads to a larger capacitance. An electrode with a big area could hold a larger charge,
creating a larger potential difference, resulting in a larger capacitance. We predicted the
capacitance of the parallel-plate capacitor would decrease if the separation between the plates
increased, which was proven via our results. For each size parallel-plate capacitor, we observed
as separation increased, capacitance decreased.
𝐴
The graph C vs. 𝑑, (figure 2), was a straight line that is equal to ε0, also proving that the
relationship between capacitance and area/separation was linear. . The experimental value of ε0
was relatively close to the literature values. A source of error could have been the disturbance of
electric field when an individual’s body part came close to the capacitor.
Conclusion:
In this experiment a measured value was calculated for the permittivity of free space constant
with the use of a parallel-plate capacitor. The calculated value of the permittivity of free space
constant was, 𝜀0 = 1 x10^-11 F/M. This was relatively close to the reported literature value,
8.85∙10-12 F/M. The mantissas varied greatly, however the exponents were close, ensuring a
somewhat accurate measurement. The slight variations could be a result of uncertainties
addressed in the analysis section.
It was concluded why there is a need for the electrostatic constant. The Permittivity of free space
constant is a physically significant quantity that is used to relate the strength of an electric field
to the electric force from a point charge.
Work Cited:
Knight, R.D., Jones, B., and Field, B. (2010). College Physics: A Strategic Approach 2nd Edition.
Illinois: Pearson Education Inc.
Selkowitz, R. Lab 2: Electric Field Mapping Handout
Delmont, Mike. Homemade Parallel-Plate Capacitor figure 1
Amir Khan
PHY 202 Lab
1.)
Lab Questions
Dr. Selkowitz’s criterion is fully met in the abstract. An abstract is meant to explain
the main idea and events in the paper, and is to be very short and to the point (one or two
paragraphs long). The hypothesis and methods of distinguishing between the two species
were introduced after giving details on the species. The percentage of accuracy in
discrimination between the two species was increased as the experimental results were
proved. This was a requirement of the criteria (qualitatively stating what the conclusion of
the experiment is through the results). That being said however, the only flaw in following
the criteria was that the abstract was not in a smaller font compared to the rest of the
paper.
2.)
The hypothesis was discussed in the abstract section of the paper. The hypothesis
of the paper stated that the shapes differ of the two species (surstyli and aculei). Also it was
hypothesized that relating the aculeus shape and size enhanced the differentiation,
(technically there were two hypotheses in this paper).
3.
The data was collected from hawthorn fruit or infested apples and snowberrys in
Washington state or northwestern Oregon in the years 2007 and 2008. Most hawthorns
and snowberries were received from the plant, whereas the apples were predominantly
gathered from the ground. The tests on the fruits were done in either Washington State
University laboratories or USDA-ARS laboratory. Next the puparia were collected
periodically and stored in soil cups at 3-4 degrees Celsius. They were stored for a minimum
of 6 months, and eventually were transferred for adult emergence at 21-27 degrees Celsius.
Finally the flies sclerotized, and were either frozen or died in the cages.
4.)
The CVA addressed the hypothesis, shown in figure four (the scatter plot). The
results of figure four revealed a strong separation in the surstylus shapes between species,
and none for the groups within species. Figure nine and figure 12 also reveled significant
results. Figure nine showed little separation in aculeus shapes of the two species. Figure 12
revealed the differences in aculeus lengths between the two species. That being said
however figure four showed the strongest separation; therefore it addressed the
hypothesis the best.
5.)
The hypothesis was indeed proved. This was at display when looking at the results
and at the figure. Figure four provided significant results for the difference in shapes
between the two species (P < 0.0001). Another crucial figure that displayed significant
results was figure nine. This figure had substantial findings for the difference in aculeus
shapes between the two species (P < 0.0001). The results support the hypothesis of the
study. Relating the aculeus’s shape and size, enhanced differentiation, and was proved as
the classification accuracy improved to 94.5%, (originally at 85.3%).
6.
The apple maggot, (R.pomonella), is a very harmful pest, that damages apples, and is
a big concern for apple industries. Also (R.zephyria) a species which resembles R.pomonella,
however is not a threat to apples. This causes major problems, due to the difficulty of
classifying between the two species, which hinders the ability to appropriately protect the
apple trees. This causes such a problem that there are tremendously expensive methods
attempting to distinguish between the two species. Such tactics include scanning electron
micrographs and genital morphology. This is a huge concern due to all the unnecessary
expenses used to properly classify the correct species, which is not as efficient as having a
non-expensive method. The ideal situation would be reducing the cost to identify the
correct species, and saving more trees from the apple maggots.