Parallel Plate Capacitors Purpose: The goal of this experiment is to determine the dielectric constant of the insulating washers that we will use to separate the parallel plates of our capacitor. A full data analysis, including the uncertainty analysis will be due. The uncertainty analysis you will need to do will involve a regression calculation similar to what you did to determine the electric field in the Electrostatic Deflection of Electrons experiment a couple of weeks ago. You will also need to do a Propagation of Uncertainties calculation similar to what you did in the “Uncertainties in Measurements” experiment in PH-2010 where we discussed “Random and Systematic Errors and Uncertainties”. Prelab exercises to be handed in as you arrive for class. a. In a “free fall” experiment, just like what you did in PH-2010, a regression of velocity vs. time produced the following results in units of cm and seconds: Regression Statistics Multiple R R Square 0.999986735 0.99997347 Adjusted R Square 0.999972487 Standard Error 0.726855182 Observations 29 ANOVA df Regression SS MS F 1 537659.7842 537659.7842 1017681.245 Residual 27 14.26459832 0.528318456 Total 28 537674.0488 Coefficients Standard Error Intercept X Variable 1 t Stat P-value Significance F 2.52778E-63 Lower 95% Upper 95% Lower 95.0% Upper 95.0% -29.91743579 0.277083637 -107.9725821 3.91822E-37 -30.48596445 -29.34890714 -30.48596445 -29.34890714 -976.465735 1.367945985 -1008.801886 2.52778E-63 -978.4517961 -974.4796739 -978.4517961 -974.4796739 What was the experimentally determined value of the acceleration due to gravity, g  g ? Give answer with units and the appropriate number of significant figures. b. The length (L) and width (W) of the sides of a rectangle are determined to be L = 3.417  0.057 cm and W = 7.933  0.094 cm , respectively. Calculate the area A  A . Use “propagation of errors” to determine A. Do not just calculate a maximum and minimum value for A. Again, make sure to give answer with units and the appropriate number of significant figure. 1 There may also be a quiz at the beginning of class on the interpretation of the regression output and about the “propagation of errors”. Equipment List: Two metal plates A container of at least 50 insulating washers Capacitance meter 12 inch ruler Vernier calipers The capacitance of two parallel plates is given by: C = K 0 where A d (1) K = the dielectric constant of the material between the plates ε 0 = the permittivity of free space, 8.854 10 -12 F/m A = the area that the plates overlap, and d = the plate separation. We will use equation 1 in many different ways in this experiment. Zeroing the meter: • Before measuring anything with the capacitance meter, you need to zero it. Zero the meter on the most sensitive scale, with the leads attached to the meter, but not connected to anything. • Check the zero of the meter periodically throughout this experiment. The Experiment You will determine the dielectric constant of the insulating washers. You will also determine an experimental value for the permittivity of free space,  0 . Variable fraction of the space filled with a dielectric Place 5 of the insulating washers between the capacitor plates in a pattern as shown in figure 1. 2 Figure 1 Part of the area of the parallel plate covered by dielectric washers. We can think of the area of the plates that is covered by washers as many parallel plate capacitors with a dielectric that are in parallel with each other and with the portion of the plate area that is not covered by washers. For capacitors in parallel we can write C = C1 + C2 + C3 + ..... (2) If we define AWasher to be the area of one washer and A to be the total area of the plates we can, using equations 1 and 2, write: C = 0 A − NAWasher NA + K  0 Washer d d (3) where N = the number of washers used. A simple rearrangement of the above equation gives: C = ( K − 1)  0 AWasher A N + 0 d d (4) This equation suggests that, if we measure C as a function of the number of washers between the plates, then a graph of C vs. N would plot as a straight line with a slope m: m = ( K − 1)  0 and an intercept b given by 3 AWasher d (5) b = 0 A d (6) The intercept correspond to if we could keep the plated a distance d apart, without using any dielectric washers. The slope is the additional capacitance we get for each added dielectric washer. Thus, if we measure the capacitance of our capacitor as a function of the number of washers, we ought to be able to determine o from the intercept and the dielectric constant of the washers from the slope of a graph of C vs. N. • With a washer close to each of the corners, and one at the center of the bottom metal plate, as shown in figure 1, add the top plate, weight it down at the center with a calculator, and measure the capacitance. • Add 5 more washers at the time, spread uniformly, replacing the calculator each time, and repeat the measurement until you have used 50 washers. • Graph the capacitance, in nanofarads [nF], vs. the number of washers and determine the permittivity of free space, o, from the intercept and eq. 6. • Determine the dielectric constant from the slope, your experimental value of o and eq. 5. • Use “Propagation of Errors” and equation 6 to estimate the overall uncertainties in  0 . To do that, you first need to determine the uncertainty in everything that goes into eq. 6. o Use Excel to do a regression calculation for your data to get the uncertainty in the slope and the intercept. o Measure the thickness of 10 stacks of 5 washers. Divide each measurement by 5 and calculate the average thickness and the standard deviation. ▪ The uncertainty in the average is the sample standard dev. of the average: s s d = savg = = # independent measurements 10 o Measure the length and width of each plate 10 times (40 measurements) ▪ Calculate the average length and width of each plate. ▪ Determine the uncertainty in each length and width. ▪ Calculate the area and the uncertainty in the area of each plate. ▪ Use the average of the two plate areas in your calculations. • Determine the uncertainty in the average plate area. o Calculate the uncertainty in  0 . o Does you value of the permittivity of free space agree with the accepted value, within your estimated uncertainty, or do you have a significant systematic error? • Use “Propagation of Errors” and equation 5 to estimate the overall uncertainties in (K-1), and then the uncertainty in the dielectric constant 4 o You already have the uncertainties in the slope, in d and in  0 . o Measure the inner and outer diameters of 10 different washers. Note – the caliper has a pair of prongs for measuring the inside diameter of a hole. ▪ Calculate the area of each of the 10 washers. Calculate the average and the standard deviation. o You do not have an accepted value for the dielectric constant. Just state your result clearly as K  K A better way to determine the dielectric constant: It may at first appear that it would be better to use the accepted value for  0 to calculate the dielectric constant, but we can actually get away from this question completely. Looking at equations 5 and 6 we see that  0 /d appear in both equations. Solving eq. 6 for  0 /d and substituting this into eq. 5 we get m = ( K − 1) b AWasher A (7) Note – you don’t need to know the value of  0 or the thickness of the washers to determine the dielectric constant. Can you explain why that is? Use eq. 7 to determine K  K . You should get EXACTLY the same value for K but a smaller uncertainty than what you got from eq. 5. Systematic errors So far, you have only estimated the possible random errors in your measurements and results. Systematic errors are often at least as important, and often much harder to determine. In this case, we have a simple way of estimating the systematic errors: Did your experimental value for  0 agree with the accepted value, within your estimated random uncertainty? If so, you can assume that the systematic errors in K as well as in  0 are smaller than your random uncertainties, and can be neglected. But, if your experimental value for  0 did not agree with the accepted value, within your estimated random uncertainty, you must then conclude that there are significant systematic errors in your results. If so, estimate the systematic errors by the percentage error between your experimental result for ε 0 and the accepted value. You can estimate the systematic error in (K-1) as the same percentage systematic error as in  0 . Use this when determining the overall 5 uncertainty (random + systematic) in your final value of the dielectric constant. State your result as K  Ktotal where K total = K random + K systematic Be sure to include the following in your informal report. • Clearly show ALL your data in table and graph form, following all the rules for tables and graphs. o Your tables should include calculated averages and standard deviations and uncertainties, as appropriate. • From the regression analysis, only include the values that you will use: m, m, b and b o Summarize your data by putting the regression values in a small table with all the other averages and uncertainties that you will need for your calculations. • Show your data analysis so it is easy to follow. This can be hand written. o Make sure to include the calculations of the dielectric constant from eq. 5 as well as from eq. 7. o Think of this as one big homework problem or a test problem. o Follow all the rules for presenting a solution on a test. o Clearly state your final result K  K including possible systematic errors. 6 Width [mm] 15,21 15,22 15,20 15,21 15,23 15,19 15,20 15,21 15,20 15,22 15,21 Bottom Plate Length [mm] 20,00 19,99 19,98 20,00 19,99 19,98 19,97 19,99 20,00 19,98 19,99 AREA 303,98 Width [mm] 15,20 15,21 15,21 15,22 15,19 15,21 15,20 15,23 15,21 15,20 Washers Outer [mm] 15,79 15,78 15,78 15,74 15,77 15,80 15,78 15,79 15,81 15,82 Inner [mm] 6,60 6,56 6,56 6,58 6,59 6,61 6,58 6,58 6,61 6,59 Area [mm^2] 161,61 161,77 161,77 160,58 161,21 161,75 161,57 161,81 162,00 162,46 thickness [mm] 0,77 0,78 0,77 0,78 0,77 0,77 0,78 0,77 0,78 0,77 3,85 3,91 3,87 3,89 3,87 3,85 3,89 3,86 3,89 3,86 15,21 15,79 6,59 161,65 0,77 3,87 Capacitance with Washers 8,93764E-14 8,93764E-12 Capacitance [nF] Top Plate Length [mm] 19,95 19,96 19,98 19,99 19,98 19,99 19,97 19,96 19,97 19,96 AVERAGES 19,97 AREA 303,74 0,5 0,45 0,4 0,35 0,3 0,25 0,2 0,15 0,1 0,05 0 y = 0,0024x + 0,3527 R² = 0,9916 0 10 20 30 Washer Count 40 EXPERIMENT Washer Count Capacitor [nF] 5 0,369 10 0,375 15 0,382 20 0,398 25 0,412 30 0,429 35 0,437 40 0,448 45 0,458 50 0,47 50 60
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