Physics II Lab E3 PDF

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TA: Bryan Rachmilowitz E3: Parallel-Plate Capacitor Part I-Capacitance of the Electrometer Table 1A: Calculating the Internal Capacitance of the Electrometer and its Cables, 30 V (Trial 1) First Voltage (V)

Second Voltage (V)

Cm (pF)

30

22

120

22

19

52.1

19

12

192.5

12

10

66

10

8

82.5 Average: N/A**

The C0 used was 330 pF. **Trial 1 data was not used in comparing with the expected value because the voltages and calculated Cm were very off and did not make sense. Sleeves were not rolled up so residual charges could have played a role in data. Table 1B: Calculating the Internal Capacitance of the Electrometer and its Cables, 30 V (Trial 2) First Voltage (V)

Second Voltage (V)

Cm (pF)

30

22

120

22

16

123.75

16

12

110

12

9

110

9

6

165 Average: 115.9

The C0 used was 330 pF. Sample Calculations from Table 1B (Trial 2): (30−22) Cm1:= (330)= 120 pF 22

4) The capacitance of the electrometer is 27 pF and the capacitance of the cables is 83 pF. This means the value of Cm is expected to be 110 ± 1 pF. The values of Cm obtained from the first trial were much higher than the expected Cm. This data was discarded and another set of data was obtained. The second set of data resulted in a mean Cm value of 115.9 pF (this average value was calculated without the fifth Cm value. The observed Cm value is not within experimental error of the expected Cm value. This is likely due to error in the measuring of electrical current. Our sleeves and the charges we produce when we moved our arms likely impacted the readings. 5) Some of our measurements did agree with the expected value within experimental errors (Cm=110 ± 1 pF). We had two measurements that agreed with the expected value which was 110 pF. Then, we took an average of all of the internal capacitance which resulted in 115.9 pF. (Note: 165 pF was taken out of the calculation for the average because it was far off than the rest. This could be due to residual charges when the person connecting the red lead of the electrometer starts shifting around.) This was close to the expected value but not within experimental errors. Part II: Fixed Charges FOR 10V: Table 2: Voltage Reading with Increasing Distance of between Parallel-Plate Capacitor at 10V Trial

Distance (mm)

Voltage (V)

V1

2

10

V2

3

11

V3

4

12

V4

5

13

V5

6

14

V6

7

15

V7

8

15

V8

9

16

V9

10

16

V10

11

16

V11

12

16

V12

13

17

V13

14

17

V14

15

18

V15

16

19

V16

17

18

V17

18

18

V18

19

18

V19

20

18

Table 3: Inverse Voltage and Inverse Separation, 10 V (Manipulated data from above)

Graph 1: Inverse Voltage vs Inverse Separation, 10 V

Observation: Voltage and distance are proportional so because of this it is getting bigger. The intercept was 0.027288 and the slope was 0.00012761. The R2 value was 0.94068. The R2 value was not as close to one but this could be due to some outliers, such as the points between 200300 m and at 500 m. This could be due to experimental error such as breathing on the plates since lab partner had to get close to the plates in order to move it accurately by 1 mm. When we divided slope by intercept, we got 0.002. This will be used later on to compare other measurements. The voltage values increases as the separation increases due to the fringing effect when area is similar to the distance. 7) To determine the experimental error, the voltage for the same distance between the plates was recorded, then the plates were moved apart and brought back to the original distance and the voltage was recorded again. A total of three observed voltages were recorded. The average of these three voltages were then determined and used to figure out the experimental error associated with the calculations from part 2. The three voltages recorded were 19, 20 and 20. 19 + 20 + 20 Average voltage : = 19.67 3

Experimental error : 19.67 - 19 = 0.67 or 4.02% Sample Calculation: --------------------------------------------------------------------------------------------------------------------Theoretically using Cm= 110 x10-12 F: q= Cm / i (110 x 10−12 F ) =4.03E-9 C q= 0.027288 A=sCm / iε0 A=

0.00012761(110 E−12) =0.058 m2 0.027288 (8.854 E−12 )

*Calculate area by geometry: R=10 ± 0.1 cm → 0.1 ± 0.001 m C=ε0A/d A=Cd/ε0 (110 x 10 −12 F )(0.2 ) =2.48 m ± 0.001 m2 A= 8.854 E−12 --------------------------------------------------------------------------------------------------------------------Experimentally using Cm= 115.9 x10-12 F: (115 x 10−12 F ) q= =4.21E-9 C 0.027288

A=

0.00012761(115 E−12) =0.061m 0.027288 (8.854 E−12 )

*Calculate area by geometry: R=10 ± 0.1 cm → 0.1 ± 0.001 m (115 x 10 −12 F )(0.2 ) A= =2.60 m ± 0.001 m2 8.854 E−12 --------------------------------------------------------------------------------------------------------------------Comparing the areas calculated experimentally with those calculated theoretically, the areas are within experimental error with one another. The experimental error is ± 4.02%. 0.058m2 and 0.061m2 are within experimental error.

FOR 15 V: Table 4: Inverse Voltage and Inverse Separation, 15 V

Graph 2: Inverse Voltage vs Inverse Separation, 15 V

Observations: Linear line was produced and few outliers. There was one outlier in which its voltage was 0.041667. This voltage was higher than those surrounding it which were around 0.037. This could have been an experimental error such as breathing on the instrument since we had to get close in increasing the distance. The R2 value was closer to 1 than previously which meant that we minimized error better. We divided slope by intercept which got us 0.002. This is the same as the value beforehand with q=10V. The voltage values increases as the separation increases due to the fringing effect when area is similar to the distance. Experimental error : 19.67 - 19 = 0.67 or 4.02% (from above) Sample Calculation: --------------------------------------------------------------------------------------------------------------------Theoretically using Cm= 110 x10-12 F: q=

(110 x 10−12 F ) = 3.28E-9 C 0.033558

A=

7.0783 E−5(110 E−12) = 0.026 m2 0.033558 (8.854 E−12 )

*Calculate area by geometry: R=10 ± 0.1 cm → 0.1 ± 0.001 m (110 x 10 −12 F )(0.2 ) A= =2.48 m ± 0.001 m2 8.854 E−12 --------------------------------------------------------------------------------------------------------------------Experimentally using Cm= 115.9 x10-12 F: (115 x 10−12 F ) q= = 3.43E-9 C 0.033558

A=

7.0783 E−5(115 E−12) = 0.027 m 0.033558 (8.854 E−12 )

*Calculate area by geometry: R=10 ± 0.1 cm → 0.1 ± 0.001 m (115 x 10 −12 F )(0.2 ) =2.60 m ± 0.001 m2 A= 8.854 E−12

--------------------------------------------------------------------------------------------------------------------Comparing the areas calculated experimentally with those calculated theoretically, the areas are within experimental error with one another. The experimental error is ± 4.02%. 0.027m2 and 0.026m2 are within experimental error.

FOR 30V: Table 5: Inverse Voltage and Inverse Separation, 30 V

Graph 3: Inverse Voltage vs Inverse Separation, 30 V

Observations: Linear line was produced and few outliers. There was one outlier in which its voltage was 0.0151515. This could have been an experimental error such as breathing on the instrument since we had to get close in increasing the distance. The R2 value was less than 1 significantly. This meant that we had a many errors in our data. We divided slope by intercept which got us 0.005. This was far off than the other results we got above. This could be due to errors in our data. The voltage values increases as the separation increases due to the fringing effect when area is similar to the distance. Experimental error : 19.67 - 19 = 0.67 or 4.02% (see above)

Sample Calculation: --------------------------------------------------------------------------------------------------------------------Theoretically using Cm= 110 x10-12 F: q=

(110 x 10−12 F ) = 1.41E-8 C 0.0078

A=

(4 E−5 )(110 E−12) =0.064 m2 (0.0078)(8.854 E−12)

*Calculate area by geometry: R=10 ± 0.1 cm → 0.1 ± 0.001 m (110 x 10 −12 F )(0.2 ) =2.48 m ± 0.001 m2 A= 8.854 E−12 --------------------------------------------------------------------------------------------------------------------Experimentally using Cm= 115.9 x10-12 F: (115 x 10−12 F ) q= = 1.47E-8 C (0.0078)

A=

(4 E−5 )(115 E−12) = 0.067 m2 (0.0078)(8.854 E−12)

*Calculate area by geometry: R=10 ± 0.1 cm → 0.1 ± 0.001 m (115 x 10 −12 F )(0.2 ) A= =2.60 m ± 0.001 m2 8.854 E−12 --------------------------------------------------------------------------------------------------------------------Comparing the areas calculated experimentally with those calculated theoretically, the areas are within experimental error with one another. The experimental error is ± 0.67%. 0.064m2 and 0.067m2 are within experimental error.

Part III: Fixed Separation: 9) Chose d=3 mm. Sample Calculation: Cp=ε0A/d Cp=ε0(�r2)/d Cp=ε0(�(0.1)2)/0.003 → 9.27E-11 qp=q

Cp C p+ C m

For Cm=115.9E-12: qp=10

9.27 E−11 9.27 E−11+115.9 E−12

For Cm=110E-12: 9.27 E−11 qp=10 9.27 E−11+110 E−12 --------------------------------------------------------------------------------------------------------------------Table 6: qp vs Vm for Cm=115.9E-12

Graph 4: qp vs Vm for Cm=115E-12

Table 7: qp vs Vm for Cm=110E-12

Graph 5: qp vs Vm for Cm=110E-12

For each of these graphs with theoretical and experimental Cm, the plot was a straight line. This means that the data is consistent with the relationship qp=CpVm. --------------------------------------------------------------------------------------------------------------------Graph 6: Plot of Total Charge q vs Vm:

Part IV: Capacitor with a Dielectric Table 8: Separation (0.0025 m) and Voltage with Foam Board, 20 V Trial 1: 0.0025 m

Voltage

With Board

20 V

Without Board

64 V

Trial 2: 0.0025 m

Voltage

With Board

20 V

Without Board

44 V

Trial 3: 0.0025 m

Voltage

With Board

20 V

Without Board

35 V

Multiple trials had to be done due to error. In Trial 1 and 2, there was a significant increase of voltage after the foam board was removed. This indicated that rubbing happened between the person and the foam board. Because of this, multiple trials had to be done. Trial 3 had the best outcome. 12) C1=ε0A/d

Cp=ε0(�(.1)2)/(0.0025) C1=1.112E-10 (1.112 E−10+110 E−12 )(35 )−(110 E−12)( 20 ) = 2.491 (1.112 E−10 )( 20 ) Our calculated K value is greater than 1 in this case. k=

Report: 1) From part III, it is concluded that charge and voltage on a capacitor are proportional to each other due to the following equation: qp=CpVm.. It was also concluded from the plot (Graph 4 and Graph 5). This means that the capacitance is a well-defined quantity and that it depends on geometrical factors. By increasing the distance, it was seen that this affected the capacitance as long as the area. It does confirm the specific expression for the capacitance of a parallel-plate capacitor. 2) Based on Parts II and III, it was reasonable to treat the capacitor as an ideal parallel-plate capacitor. The capacitance of a parallel-plate capacitor whose plates are conducting A squares of side a placed a distance d apart is given by C=ε o . This can be d exemplified by the data from Part II in which the Area of the capacitor could be derived 1 1 from a plot of V m vs. d using the assumption that the parallel plate capacitor was ideal. The calculated area for all trials agreed within experimental error of the expected value that was calculated. It is only reasonable to treat the capacitor as an ideal parallel plate capacitor at small distances. 3) For Part IV, the result for the dielectric constant was reasonable. It was slightly lower than for a polystyrene, k=2.6, but it was by 0.11. This could have been due to error such as rubbing which caused the change of voltages being too significant. The capacitance will increase when a dielectric material is inserted between a capacitor’s plates. This is because k>1. When a dielectric material is added, it reduces electric fields. Also, it pushes more charges onto the capacitor since there are more opposite charges near the like charges....