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Parallel-Plate Capacitor Measuring the Value of the Permittivity of Free Space (๐0 ) 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 ๐0 , was calculated through the knowledge of the relationship between ๐0 and our measured variables. The Slope ๐ด obtained from the graph of C vs. was determined to be the calculated value of ๐0. The experimental value of ๐0 ๐ 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 (๐0 ) is used to relate units of electrical charge to mechanical quantities (area and length). To calculate the numerical value ๐0, 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 ๐0 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, โthe constant of proportionality C between charge on the capacitor and the potential difference between the two plates, the equation is given below ๐ C=๐ฅ๐ , (1) ๐ถ Where C is the capacitance (in farads), Q is the magnitude of charge (in Coulombs) on the plates, and ๐ฅ๐๐ถ 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: ๐ Ecapacitor = ๐ ๐ด, (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 ๐0 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= ๐ฅ๐๐ถ ๐ , (3) The equation then becomes: Q= ๐0 ๐ด ๐ ๐ฅ๐๐ถ, (4) When equation 1 is compared to equation 4, we can determine the parallel plates capacitance as C= ๐0 ๐ด ๐ , (5) Where C is the capacitance (in Farads), ๐0 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 ฮต0 ฮต0 = ๐ถ๐ด ๐ , (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 โsmallโ boardโs length was measured with a meter stick, and the area was calculated, (process was repeated for the other two sized boards). Using the โsmallโ 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 โmediumโ and โlargeโ 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. ๐ด/๐ 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. ๐ด The graph C vs. ๐, (figure 2), was a straight line that is equal to ฮต0, also proving that the relationship between capacitance and area/separation was linear. . The experimental value of ฮต0 was relatively close to the literature values. A source of error could have been the disturbance of electric field when an individualโs 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, ๐0 = 1 x10^-11 F/M. This was relatively close to the reported literature value, 8.85โ10-12 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. Selkowitzโs 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 aculeusโs 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.