Measuring Permittivity of Free Space

Etudes

Measuring Permittivity of Free Space: ?0
Daisy Mota
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Septermber 25, 2018Lab Partner: Lauren Shephard1000000
Septermber 25, 2018Lab Partner: Lauren Shephard

Permittivity of Free Space: Determine Epsilon naught
Purpose: Through excel modeling an experimental value for the fundamental constant ?0= 8.854×10-12F/m is calculated by graphing capacitance vs 1/distance.

Procedure:
171452984500013335180011left888365Figure 1
00Figure 1
This experiment uses parallel plate capacitors, spacers, capacitance and calipers in order to calculate Epsilon naught. The spacers will be used to create distance between the parallel plate capacitors.
right1583690Figure 2
00Figure 2
4577715154178000Using the caliper, measure the thickness of the spacer by inserting the spacer between the teeth of the caliper. Before taking the spacer measurement, ensure that when the teeth are closed the screen reads zero. Measuring the spacers is illustrated in figure 1. The spacers will measure approximately 2 cm/.002 m. Now measure the length and the width of the parallel plate since the area will be needed in the calculations later. Next place spacers at the corners of one plate so that only one quarter of the circle spacer is touching the plate. Once there is one spacer at each of the corners of the parallel plate, place the second parallel plate on top of the first while being carefully not to misalign the spacers. The parallel plates should be placed in such a way they resemble figure 2. Next, ensure the capacitance meter is zeroed by gently holding the tips of the wire in the plate ports (not inserted) and manually turning the capacitance meter small dial to zero. The capacitance meter only needs to be zeroed initially.
-95250825507677151093470Figure 3
00Figure 3
63500869950007702200124241512058966838950011982436209690010096561404500901706629410067437087249000679450810895000-190500After zeroing the meter, the wires can now be inserted into the ports to measure the capacitance figure 3. Turn the main dial of the capacitance meter to 200pF in order to measure capacitance. Record the measurement that appears on screen (keep in mind the conversion from pF to F). Proceed to repeat this measurement process with an increasing amount of spacers/distance. After completing the capacitance reading for a 6 spacer distance, graph the capacitance along the y-axis and 1distance along the x axis in Excel. From the resultant graph, ?0 can be calculated through the slope and the area of the parallel plate.
Data:
Area of parallel plate = (.175m)(.175m)=.0306m2
Number of Spacers Separation/d
(cm) Separation/d
(m) 1/d (m-1) Capacitance (F)
1 .2 .002 500 1.37×10-10
2 .4 .004 250 6.92×10-11
3 .6 .006 166.67 4.49×10-11
4 .8 .008 125 3.31 x10-11
5 1.0 .010 100 2.54 x10-11
6 1.2 .012 83.3 2.04 x10-11
Theory:
Capacitance of a parallel plate is calculated by multiplying the constant ?0 by the Area then dividing by the separation(d) of the plates. This is derived by the definition of capacitance.
C=Q?V302895057150-?
00-?
245110052070?
00?
3136900124460002565400132080
3352800106680iii
00iii
2755900100330ii
00ii
219075093980i00iEii= ?2?o+?2?o=??o
?V is equal to ?V= -0d Edl , and in the case of parallel plates, the E is equal to ??o= Q?oA. Next take the integral.

?V= -0d Q?oAdl
?V=Q?oAdNow substitute ?V back into the capacitance formula C=Q?VC=QQ?oAdAfter cancellation (of Q), our capacitance formula for a parallel plate is:
C= ?o*Areaseperation=?o*AdIn order to experimentally calculated ?0, simply graph Capacitance against the inverse of separation to determine our slope. In the general form of a linear equation, y=mx+b, our “y” will represent capacitance, our “m” will be our (?o*A) and our “x” will be the separation(d). Once the slope has been calculated, the Area of the plates can be divided to find our constant!
C= ?o*Areaseperation=?o*AdC= ?0*A * 1d y=mx
Slope= ?0*Ayx =m
?0= slope areaAnalysis:
right20199
1/d (m-1) Capacitance (F)
500 1.37×10-10
250 6.92×10-11
166.67 4.49×10-11
125 3.31 x10-11
100 2.54 x10-11
83.3 2.04 x10-11
A= length*widthA= .175m*.175m=.0306m2
Slope= ?0*A
?0= slope area?0= 2.80×10-13F*m .0306 m^2 = 9.150×10-12 F/mError %:
% error = measured-theorecticaltheorecticalx100% % error = 9.150×10^-12 -8.85×10^-12 8.85×10^-12 x100% = 3.39%
Results:
Our calculated ?0= 9.150×10-12 F/m with a percent error of 3.39%.
?0= 2.80×10-13F*m .0306 m^2 = 9.150×10-12 F/m
% error = 9.150×10^-12 -8.85×10^-12 8.85×10^-12 x100% = 3.39%
Error Analysis:
Capacitance readings are affected by our bodies being near the capacitance meter and when we touched the parallel plate when adding more spacers. Error may have been introduced through the spacers. Since the spacers have a non-trivial dielectric constant if there was a lot of contact with the spacers and the plates it could skew the results. Finally our results could have been affected by the measuring equipment used in the experiment such as the capacitance meter and the lack of precision in the ruler used to calculate the area of the plates. Conclusion:
An experimental version of epsilon naught was calculated through the capacitance equation for parallel plates. By graphing capacitance versus 1seperation the resultant slope was divided by the area for our experimental value of ?0= 9.150×10-12 F/m.
2971804699000Appendix: