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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.