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Determining the Permittivity of Free Space with Parallel Plate Capacitors
Emily Gregorio
Sarah McMahon
Kyle Lambert
Abstract:
Capacitors are capable of storing and releasing charge, like a temporary battery, and can be used in circuits as an alternate
source of voltage. Here, we used rudimentary parallel-plate capacitors, made of aluminum and fiber board, of various area and
separation to determine a common constant in electrostatics, Ɛₒ. Confirming the accepted value of Ɛₒ would then allow us to do
further electrostatics problems that require this constant. To demonstrate the relationship between capacitance and spatial
variables of the systems, a graph was constructed in which the slope would represent Ɛₒ. Our value of Ɛₒ was 9.27*10-12
C2/N*m2, compared to the accepted value of 8.85*10-12 C2/N*m2, resulting in a percent error of 4.75%. Considering the low
percent error and the linear relationship of our graph, we can say the accepted value of Ɛₒ was successfully confirmed.
Introduction and Theory:
Previous experimentation suggests everyday objects accumulate excess electrical charges (Gregorio et
al, 2015a). Furthermore, it has been observed that there are two distinct and opposing types of charges,
positive and negative. This accumulation can be confirmed by placing other suspected charged or
neutral objects nearby and noting the nature of the response. This response is due to electric fields
created by the charges (Gregorio et al, 2015b). Parallel plate capacitors are an iconic example of
oppositely charged electrodes interacting in a manner that creates a constant electric field between
them.
The permittivity of free space, Ɛₒ, is a constant that appears in several formulas involving interacting
charges (Ɛₒ is a component of the electrostatic constant, K). Because Ɛₒ is a constant, it should be
possible to reproduce the numeric value in a laboratory setting. Understandably, Ɛₒ is used in many
equations for parallel-plate capacitors. A parallel-plate capacitor is an arrangement of two parallel,
oppositely charged electrodes (charged plates) through which an electric field is generated and charge
can be stored and/or released. It is important that we keep in mind previous assumptions associated
with parallel-plate capacitors, specifically those of the electric field formula,
𝑄
𝐸⃗ = ƐₒA ,
(1)
where E is the electric field, Q is the magnitude of charge on the plates, Ɛₒ is the permittivity constant,
and A is the area of the plates. This equation asserts that the electric field is larger with a smaller plate
area and greater magnitude of charge. Voltage is created by the separation of positive and negative
charges, as in a parallel-plate capacitor. The relationship between the voltage and electric field of a
capacitor is shown as
∆𝑉
𝐸⃗ = d𝑐 ,
(2)
Figure 1: Parallel plate capacitor setup; the plates are separated by wooden spacers (yellow), weighted down, and the
aluminum foil tags are attached to the black box capacitance meter; charges are arbitrarily assigned.
where ∆Vc is the potential difference (voltage) of the capacitor and d is the separation of the plates.
Since both equations represent electric field, substituting equation 2 into the value of E in equation 1,
results in
∆𝑉𝑐
d
𝑄
= ƐₒA .
(3)
Solving equation 3 for Q results in the equation
𝑄=
ƐₒA
∆𝑉𝑐
d
.
(4)
Based on the equation for charge on a capacitor,
𝑄 = 𝐶∆𝑉𝑐 ,
(5)
we can conclude that
𝐶=
ƐₒA
𝑑
,
(6)
where C is capacitance, Ɛₒ is the permittivity constant, A is the plate area, and d is the separation
between the plates. Equation 1 assumes that the field between the plates is uniform (Knight et al, 2014).
Because equation 1 was crucial to deriving equation 6, it is mandatory that this assumption is met; if it is
not met, our system is no longer true to the definition of parallel plate capacitor.
Based on equation 6, we predicted that capacitance would increase with area and decreased
separation. This prediction, while not crucial to the goal of the experiment, was necessary for identifying
any possible procedural errors during experimentation.
To determine the value of Ɛₒ, we set up a parallel-plate capacitor system using fiber board with
aluminum foil wrapping, and then measured capacitance with a black box capacitance meter (figure 1).
We hypothesized that a graph of C versus A/d would result in a directly proportional trend line, linearity
determined by an R2 value close to 1, with a slope of Ɛₒ. This is based on applying the general line
formula,
𝑦 = 𝑚𝑥 + 𝑏 ,
(7)
where y is the dependent variable, m is the slope, x is the independent variable, and b is the y-intercept,
to equation 6. We understood that capacitance will be dependent on the area and separation of plates,
so plotting these should result in a line with a slope of the constant Ɛₒ.
Procedure:
Before testing, length and widths for the three pairs of plates were measured using a meter stick.
Average separator width was then estimated using a caliper. Knowing
𝐴 = 𝑙 ∗ 𝑤,
(8)
or the area of a square (A) is equal to the length (l) multiplied by the width (w), we calculated the area of
each plate pair.
The plates were assembled with aluminum sides facing each other and five separators in between: one
per corner and one in the middle. It was important that we refrain from using too many wood
separators because wood is an insulator. Direct contact between the wood and capacitor’s plates results
in atomic polarization, or the lining up of atomic dipoles as a result of the torque exerted on charged
atomic particles by the capacitor’s applied electric field, in the wood because the electrons in the atoms
are localized (can’t move around as in the “sea of electron” model with metals/conductors). For the
system, this means that wood has a dielectric constant greater than that of air, or 1.00054. Presence of
a dielectric increases the capacitance; the capacitor’s electric field causes polarization of the dielectric,
resulting in an opposing induced field within the dielectric and thus decreasing the net electric field of
the system (Knight et al, 2014). Looking at equation 2, we see that electric field and voltage are
proportional to one another. This means decrease in electric field causes a decrease in voltage. With
equation 5, we see capacitance and voltage are inversely proportional so decreased voltage means
increased capacitance. Conversely, equation 6 shows smaller separation means increased capacitance,
so it was also necessary we refrained from using too few separators else there would be areas where
the plates are closer (top plate would sag due to its weight alone if not supported by a separator) with
higher capacitance.
Knowing separation affects the uniformity of the electric field, a weight was placed on top of the plate
setup to prevent areas of uneven spacing. The capacitance meter was then clipped onto the aluminum
foil tags extending from the plates and turned on. Capacitance readings were taken at separation
intervals of one, two, and three wood pieces wide for each plate pair (table 1). We then calculated the
values for A/d for each separation interval with each plate.
Plates
d (m)
0.0022
0.0044
0.0066
0.0022
0.0044
0.0066
0.0022
0.0044
0.0066
Small
(0.1515 x
0.1515 m)
Medium
(0.1970 x
0.1970 m)
Large
(0.3015 x
0.3015 m)
A (m²)
0.022952
0.022952
0.022952
0.038809
0.038809
0.038809
0.090902
0.090902
0.090902
C (F)
9.62E-11
6.61E-11
3.83E-11
1.68E-10
7.84E-11
5.64E-11
3.9E-10
1.94E-10
1.37E-10
A/d (m)
10.43284
5.21642
3.477614
17.64045
8.820227
5.880152
41.3192
20.6596
13.77307
Table 1: Data collected from each plate system; d is separation in meters, A is plate area in square meters, and C is capacitance
in Farads.
Effects on Capacitance with Spatial Variables for
a Parallel Plate Capacitor
4.5E-10
4E-10
3.5E-10
C (F)
3E-10
2.5E-10
2E-10
1.5E-10
1E-10
5E-11
0
0
5
10
15
20
25
30
35
40
45
A/d (m)
Figure 2: Compilation of capacitance and corresponding A/d for all plates; y = 9.27E-12x + 5.02E-12; R² = .997
Analysis and Results:
All capacitance and corresponding A/d values were graphed and fitted with a linear trend line (Figure 2).
The R2 value is reasonably close to 1, indicating a linear proportionality to C and A/d. The standard value
of Ɛₒ is 8.85*10-12 C2/N*m2 and our slope value is 9.27*10-12. Using the error formula
𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝐸𝑟𝑟𝑜𝑟 =
|𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝑉𝑎𝑙𝑢𝑒−𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑉𝑎𝑙𝑢𝑒|
∗
𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝑉𝑎𝑙𝑢𝑒
100 , (9)
we can see that our value of Ɛₒ has a percent error of 4.75%, which is relatively low given the primitive
nature of the experimental setup. A notable concern with our graph is the wide spacing of the points,
especially the one furthest at x=41(m). This point in particular is with the largest area and small spacing.
It is likely that portions of the top plate were sagging closer to the bottom as a result of large area (there
were large spaces where there is no separator to support weight of the plate). As we noticed by
equation 6, closer proximity means larger capacitance, thus this plate would have regions of higher
capacitance, which showed up in our graph. Two possible solutions to minimize large gaps in points
would be to either use plate areas that aren’t so drastically different in area, or to use plates of
intermediate values. If large area was in fact causing areas of increased capacitance, using smaller plates
altogether would help decrease the point spread.
It is likely that the dielectric nature of the wood separators increased the capacitance values, leading to
the presence of a y-intercept. This inconsistency may also be a significant contributor to the deviation
between the experimental Ɛₒ value and the theoretical value. Other possible sources of error include the
measurement of the wooden spacers and/or the measurement of the plate. The possibility of
measurement error for the wooden spacers is considerably small as a caliper was used to measure their
width. Calipers are very precise tools and we took several measurements to ensure we observed the
same value consistently. Measurement of the plate area could also have been a source of concern as we
used meter sticks, which are less precise and are prone to user error (such as misinterpret lines or
estimating in-between values wrong).
A reoccurring issue we experienced was having the system short towards the ends of the plates; pieces
of the aluminum would flop down and touch the bottom plate. These areas, if not touching, would be
sources of closer proximity. By equation 6, decreased separation increases the capacitance, so this may
also be a reason for inconsistency in Ɛₒ. Furthermore, there were some tears and small gaps in the foil.
This could also be a source for error.
Conclusion:
Both the value for permittivity of free space and the proportionality of capacitance to plate area and
separation have been sufficiently supported. Some shortcomings of this experiment were that we only
tested square plates with aluminum, capacitances were low, and the system was not in a vacuum. In the
future, further experimentation could test how changing the plate shape/metal type/magnitude of
capacitance affects the relationship in equation. Further experimentation could also explore the
relationship between capacitance and various dielectric constants used.
Previously, we worked with point sources to establish a generalization of how charges interact with each
other (Gregorio et al, 2015a). Specifically, we worked with Coulombs Law,
𝐹1 𝑜𝑛 2 = 𝐹2 𝑜𝑛 1 =
𝐾|𝑞1 ||𝑞2 |
𝑟2
,
(10)
where F is force, K is the electrostatic constant, q represents some point charge, and r is the separation
between the charges. We then looked at how forces are created by electric fields, establishing a
relationship between field lines and distance from point charges (Gregorio et al, 2015b). This helped
establish the equation,
𝐾|𝑞|
𝐸⃗ = 𝑟2 ,
(11)
where E is the electric field from a point source, q, at a distance, r. Knowing
𝐾=
1
4𝜋Ɛₒ
,
(12)
this experiment has effectively completed the exploration/confirmation of Coulomb’s Law. Having a
value for Ɛₒ allows for the use of other equations as well.
While confirming Ɛₒ has allowed us to connect the relationship between mechanical quantities (force
and distance) to electrical quantities, this experiment has also provided a basic understanding of parallel
plate capacitors. With this information, we will be able to delve further into the application of capacitors
and their function(s) in a larger system, such as a circuit.
Works Cited:
Gregorio, Emily, Sarah McMahon, and Kyle Lambert. Experiment 2: Electric Charges. 2015.
Gregorio, Emily, Sarah McMahon, and Kyle Lambert. Experiment 3: Electric Field Mapping. 2015.
Knight, Randall D., Brian Jones, and Stuart Field. College Physics: A Strategic Approach. 3rd ed.
Pearson, 2014. Print.