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Transcript
Using a Parallel-Plate Capacitor to Test the Permittivity of Free Space
Stephanie McDonald
Leah McDonald, Myranda Empric
Department of Physics, Canisius College
Abstract:
A simple capacitance system was constructed using three different sizes of aluminum foil covered fiber
board, an AC/DC power supply and wooden popsicle sticks to use as separators between the parallel plates.
Using parallel plate capacitors the numerical value of the permittivity of free space will be measured. After
setting up the apparatus with the parallel plates separated by one piece of wooden popsicle to be used as an
insulator we turned the power supply on to see where the needle on the AMM read to give us the
capacitance value of the parallel-plate capacitor. We turned off the power supply and added popsicle pieces
to each corner to make the distance increase for each reading. We repeated this for all three sizes of plates
and recorded the capacitance values. By carrying out this experiment we were able to determine the
relationship between capacitance and the area/ distance of each pair of parallel plates. The results indicated
a strong correlation between he measured capacitance and area/ distance. The value for
was determined
-12
to be 11.08 * 10 .
Introduction and Theory:
In this experiment we will explore how voltages are distributed in capacitor
circuits in parallel combinations. Capacitors are used in electronic circuits where it is
important to store electric charge and/or energy. A capacitor is made up of any two pairs
of conductors that can be charged electrically so that one conductor has a positive charge
and the other has a negative charge. The capacitor is a measure of a device’s ability to
store a charge. They have fixed values of capacitance, negligible resistance, and are
passive electronic devices. They can be made up of two arbitrarily shaped pieces of metal
or can be made up of two symmetric shapes, such as this parallel-plate capacitor in this
experiment.
Parallel Capacitors:
In parallel the components are connected at both ends. The capacitors are
connected and electrons leave the positive plate and go to the negative plate until
equilibrium is reached. This is when the voltage of the capacitor is equal to the voltage of
the battery. The field strength for the field between two equal but opposite charged
parallel plates is E=V/d. The voltage difference between the two plates can be expressed
in terms of the work done on a positive test charge q when it moves from the positive to
the negative plate. This can be represented by the equation V=workdone/charge=
Fd/q=Ed.
Parallel-plate capacitors are perhaps the easiest to construct and analyze in a
laboratory. It has been suggested that each unique parallel-plate capacitor carries its own
capacitance, with its stored charge. Electric constant is important while discussing
parallel-plate capacitors and capacitance. This can be represented by characteristics of
these capacitors as well as the permittivity of free space:
(1)
In this experiment, we construct a simple parallel-plate capacitor, as shown in figure 1,
and use three different sizes of aluminum-covered fiber board in order to measure the
numerical value of the permittivity of free space. Small, medium, and large sizes are used
for the boards.
First we determined the predicted capacitance which can be found by using:
(2)
Q is the magnitude of charge stored on each plate and V is the voltage applied to the plates. Capacitance is
measured in units of Farad (F), which is given by:
(3)
C is 1 coulomb and V is 1 volt. 1 coulomb is a large charge, so a farad can be said to be a very large unit as
well.
Values of capacitors found in typical circuits range from picofarads (pF) to microfarads.
(µF)
Figure 1: The experimental setup of a parallel-plate capacitor attached to a power supply
When using parallel plate capacitors the dimensions of each plate being used must be
taken into consideration in order to determine the capacitance. In this experiment the
parallel-plate capacitor consisted of two parallel metal plates separated by a distance d.
The capacitance of these plates at a distance can be found by using:
(4)
is the permittivity of free space and A is the cross-sectional area of the plates.
The permittivity of free space can be determined by plotting the data on a C vs. A/d
graph. By finding the area of the plates, distance separating the two plates, and the
capacitance using the power supply this can be determined.
Procedure:
First we obtained three pairs of parallel plate squares constructed from fiber board
and covered in aluminum foil, all of different lengths. We measured each square in order
to determine the area (m2) of each of the three parallel plates. The areas were determined
to be 0.225 m2, 0.392 m2,.0906 m2. We then separated each pair of the two plates using
pieces of wooden popsicle sticks with a width of .0026 m. This created the distance (d)
between the plates. Wood is an insulator which means it will have no effect on the
charged plates so it won’t affect the outcome of the experiment.
We started with the two smallest plates with the area of 0.225 m2 and put one
piece of a popsicle stick between them to separate and give a distance between. We
connected the red wire to the power supply and clipped the other end of the red wire to
the top plate. The black wire was then connected to the power supply and clipped to the
bottom plate which can be seen in figure 1.
A small weight was placed on top of the parallel-plate capacitor in order to ensure
the structure was tightly bound and that the distance separated was only from the popsicle
stick pieces. We then turned the power supply on and the needle on the AMM showed the
capacitance value of the parallel-plate capacitor. We turned off the power supply and
added another popsicle stick on top of the one already there to each corner and repeated
recording a capacitance value by turning the power supply on. We repeated this process
five times so each corner had five popsicle stick pieces when we were done. We then
repeated the whole process using the other two different-sized parallel- plates with the
areas of 0.392 m2 and .0906 m2. The largest plate we took four readings.
The data collected for each parallel plate pair was inputted onto Microsoft Excel. A graph
was created comparing C vs. A/d. The best line fit from the data provided us with our
numerical value of the permittivity of free space, .
Analysis and Results:
Small sheet: A=0.225 m2
Distance(m)
Area/Distance
(m)
Capacitance (F)
.0026
8.653846
9.95E-11
.0052
4.326923
5.43E-11
.0078
2.884615
3.71E-11
.0104
2.163462
3.12E-11
.0130
1.730769
2.53E-11
Medium sheet: A= .0392 m2
Distance(m)
Area/Distance
(m)
Capacitance (F)
.0026
15.07692
1.79E-10
.0052
7.538462
9.72E-11
.0078
5.025641
6.75E-11
.0104
3.769231
5.14E-11
.0130
3.015385
4.03E-1
Large sheet: A= .0906 m2
Distance(m)
Area/Distance
(m)
Capacitance (F)
.0026
34.84615
3.94E-10
.0052
17.42308
1.98E-10
.0078
11.61538
1.456E-10
.0104
8.711538
1.115E-10
Table 1. Results of the comparison of area of parallel plates and distance separated versus capacitance
The data from Table 1 was then graphed in a scatter plot with the line of best fit
displayed:
Figure 2. Sample data showing the capacitance versus area/distance of three different-sized parallel plates.
Note that the slope of the line is the permittivity of free space for this sample data.
The results indicate a fairly strong correlation between the predicted and measured
capacitance relationship to A/d. This provide us with our numerical value of the
permittivity of free space, the electric constant , as the slope of the line of best fit. Our
value for
was determined to be 11.08 * 10-12. This value is higher than the accepted
value shown in equation 1. The errors that may have occurred could have came from the
aluminum foil not being wrapped securely on the fiber boards, the popsicle stick pieces
having different widths, or misreading the needle on the AMM that displayed the
capacitance each time it was turned on.
Conclusion:
In this experiment we have shown the relationship between A and d being
proportional to C. We determined the value of
after having three
different parallel plate capacitor tests ran we determined our value for
to be 11.08 *
10-12. The percent error was determined to be 25.2%, all though our value was higher
than we anticipated we can still conclude that there is a relationship between the area of
each parallel plate and the distance between them when a current is ran through verse the
capacitance value that is taken when the power supply is turned on.