Experiment 16 Electric Field And Electric Potential, Lab Report PDF

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Experiment 16 Electric Field and Electric Potential Name.......... Lab Partners: Name.................................... TA: Kai Zhang October 14th, 2016 Abstract This experiment explored electric field lines across different electric potentials between two electrodes. Measurements of electric fiel...

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Experiment 16 Electric Field and Electric Potential

Name.......... Lab Partners: Name.................................... TA: Kai Zhang October 14th, 2016

Abstract This experiment explored electric field lines across different electric potentials between two electrodes. Measurements of electric field, electric potential, distance, average distance, and their errors were calculated and compared with theoretical values. The results from Investigation 1 produced an electric field E= 0.954 ± 0.0202 V/cm across the first measured electric field line. This deviated from the expected value E= 1.0 V/cm. Measurements from Investigation 2 allowed theoretical electric potential to be calculated and compared to observed electric potential. Large deviations in the results were likely caused by faulty equipment and error in determining the precise location of a particular electric potential.

Introduction This experiment investigated electric potential within an electric field. Electric potential and electric fields between two electrodes were studied. A primary purpose of this investigation was to understand the relation between electric potentials and electric fields. The electric field is a vector field which is related to the forces between charges. Each point in its space is associated with a vector. In contrast, the electric potential field is a scalar field, and is related to energy between charges. The relationship between electric field E and electric potential V is such that 𝐸" = −

∆& ∆"

,

(1)

where Δx is the change in distance. Equipotential lines connect points of the same potential, enabling for the visualization of the electric potential field. The electric field is always perpendicular to equipotential lines. In Investigation 1, electric potentials and electric field distributions of a parallel plate capacitor were examined. Two parallel electrodes were hooked up to negative or positive terminals, creating an electric field. A voltmeter was used to visualize equipotential lines at several electric potentials. Electric field and electric potential values were observed and compared to the expected values. In Investigation 2, electric field between concentric electrodes was examined. Two circular electrodes of different sizes were used concentrically to create an electric field. Measurements of the inner radius a, outer radius b, electric potentials, electric field, and distances were calculated. From this, theoretical electric potentials at each distance could be calculated. Deviation between observed and theoretical electric potentials were likely caused by large error in voltage, as well as by faulty equipment. Additional experimental and random errors likely contributed to deviation in results.

Investigation 1 Setup & Procedure In this investigation, the electric field between parallel electrodes was examined. Two parallel electrodes were placed 10.0 ± 0.1 cm apart on conducting paper on a rubber pad. The rubber pad was used to isolate the currents to the conducting paper. A grease pencil was used to mark the edges of the electrodes, guiding in the visualization of the electric fields. The power supply was then connected to the two electrodes. The negative terminal was connected to the electrode on the left, where x=0, and the positive terminal was connected the electrode on the right, where x= 10.0 ± 0.1 cm. The power supply was set to 10 V. The left, negative electrode was grounded and connected to the voltmeter. This indicated that the voltage at x=0 was 0 V, or V (x=0) = 0 V. The probe-tipped wire was connected to V input of the voltmeter, and the precision of the voltmeter was set to 20 V max. In increments of 1 V, equipotential lines were traced between the electrodes on the paper. The 5 V line was traced first. This was expected to be located about x= 5 cm in from each electrode. Gentle and equal pressure was applied to each electrode in attempt to ensure there was good contact between the electrodes and paper. After completing the equipotential lines, the power supply was turned off and disconnected.

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Data & Analysis Electric field lines between the electrodes were drawn in the center, as well as about every inch along the electrodes. The distances ∆x between neighboring equipotential lines were measured along the field line at the center of the electrodes. Thus, the change in electric potential ∆V was 2 ± 0.2 V. Using these measurements, the electric field was calculated for each pair of equipotential lines with the equation 𝐸" = −

∆& ∆"

,

(2)

where Ex was the electric field for a particular line. Plots of V vs. x and E vs. x were then made. Each E was calculated from an average field between a pair of potential lines.

Electric Potential V (V) vs. Position x (cm) y=x R² = 1

Electric Potential V (V)

10

8

6

X1

y = 0.954x + 0.0558 R² = 0.99681

X3

4

Theoretical X1

2

0 0

2

4

6

8

10

Position x (cm)

Figure 1. Plot of Electric Potential vs. Position. The electric potential at each location of two of the field lines are shown. The electric potential V is on the y-axis; position x (cm) is on the x-axis. The points in blue “x1” are the electric potential at each position along the first measured electric field line. The points in red “x3” are the electric potential at each position along the third measured electric field line. The points in purple “Theoretical x1” are the theoretical electric potential at each position along the firs measured electric field line. The line of best fit for the x1 set of data is represented by the equation y= 0.954x + 0.0558. The value of R2= 0.99681. Horizontal and vertical error bars indicate small errors. The line of best fit for the theoretical x1 data is represented by the equation y= 1x, with an R2= 1.

The average distance ∆x1 for the first measured electric field line, E1 was ∆x1= 2.186 ± 0.1421 cm. The average distance ∆x2 for the second measured electric field line, E2 was ∆x2= 2.187 ± 0.1421 cm.

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The average distance ∆x3 for the third measured electric field line, E3 was ∆x3= 2.770 ± 0.1421 cm. This measurement error 𝛿Δx was calculated by multiplying the propagated error 𝛿x by 2. The propagated error 𝛿x= 0.1 cm. This was determined based on the possible measurement error that could have occurred using the ruler used. There was likely little error that came from ruler measurements. These values were used to calculate the electric field E at each at each V(x). The slope of the line of best fit in Figure 1, 0.954, is the observed electric field E (V/cm). The expected electric field E value was 1.0 V/cm. This was calculated using the relationship V(x) = -Ex. Since V was a constant 10V at x=10cm, E is expected to constantly equal 1.0 V/cm. The observed value of E was 0.954 ± 0.0202 V/cm (Fig. 1). This error was calculated using the online error calculator. The theoretical and observed values of the electric field were very close, but were not equal within calculated errors. According to these calculations, E was expected to equal 1.0 V/cm at all measured locations. However, the observed electric field did not equal 1.0 V/cm at all locations within calculated error (Figs. 2, 3 and 4). The E observed at the first electric field line was equal to the theoretical value within experimental error (Figure 2); however, E observed at the second and third electric field lines were not consistent with what was expected within experimental error (Figs. 3 and 4). Using the relationship V(x)= -Ex, each theoretical electric potential V was then calculated by multiplying the theoretical E= 1V/cm by the measured location x. Thus, each theoretical V was equal to the corresponding location x. This relationship was visualized in Figure 1, with a relationship of y= x for the set of data in purple.

E1 (V/cm)

Electric field E1 (V/cm) vs. Position x (cm) 1.18 1.13 1.08 1.03 0.98 0.93 0.88 0.83 0.78 0.73 0.68 0.98

1.98

2.98

3.98

4.98

5.98

6.98

7.98

8.98

Position x (cm)

Figure 2. Electric field vs. Position x. E1 (V/cm) is on the y-axis; position x (cm) is on the x-axis. Horizontal and vertical error bars shown for each point.

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Electric field E2 (V/cm) vs. Position x (cm) 1.4 1.3

E2 (V/cm)

1.2 1.1 1 0.9 0.8 0.7 0.6 0

1

2

3

4

5

6

7

8

9

10

Position x (cm) Figure 3. Electric field vs. Position x. E2 (V/cm) is on the y-axis; position x (cm) is on the x-axis. Horizontal and vertical error bars shown for each point.

Electric field E3 (V/cm) vs. Position x (cm) 1.3 1.2

E3 (V/cm)

1.1 1 0.9 0.8 0.7 0.6 0

1

2

3

4

5

6

7

8

9

10

Position x (cm) Figure 4. Electric field vs. Position x. E3 (V/cm) is on the y-axis; position x (cm) is on the x-axis. Horizontal and vertical error bars shown for each point.

The error of electric potential 𝛿V was estimated based on the variation in voltage as different pressures were applied to the electrodes. However, we likely underestimated how much error there actually was in observed electric potential. Pressure was applied to each electrode in order to ensure good contact between the electrodes and the paper. Slight variations in the amount of pressure, however, caused a change in apparent electric potential at any particular spot. This significantly contributed to deviations

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in the results, suggesting that there should have been a greater incorporated error in our calculations. An additional experimental error could have contributed to deviations, as well.

Investigation 2 Setup & Procedure In the second investigation, electric fields between concentric electrodes were examined. The electric field lines were visualized on the opposite side of the same black paper that was used in Investigation 1. A pair of perpendicular lines were drawn in the center of the paper. A pair of a large and small brass ring was used as concentric electrodes. The radius a of the smaller (inner) electrode was measured (a= 1.2 ± 0.1 cm). The radius b of the larger (outer) electrode was measured, as well (b=10.8 ± 0.1 cm). The grease pencil was then used to draw an outline of the electrodes. The grounded negative terminal of the power supply was connected to the outer electrode, and the positive terminal was connected to the inner, center electrode. The power supply was set to 10V. The voltmeter ground was then attached to the outer ring, and the probe-tipped wire was connected to the V input of the voltmeter. The voltmeter probe tip was fine-tuned to ensure that the power supply voltage was 10V. Every 45°, the 5V equipotential points were marked. Then, 4V, 3V, and 2V points were marked, followed by 6V and 7V points. The power supply was then disconnected, and electrodes were removed from the paper.

Data & Analysis All of the measured points were connected in order to obtain equipotential lines. The lines were approximately circular at some points, but not at other points. The radial distance ri from the coordinate origin to the equipotential lines were measured along each of the coordinate axes. Two distance measurements, r1 and r2 were obtained for each equipotential. The average radial distance ravg, error 𝛿r (0.1 cm), and error 𝛿ravg (0.28 cm) were then calculated for each of the measured equipotential lines. The four measured equipotential lines were 3, 4, 5, and 6 V. A plot of electric potential V vs. average radius ravg was made using this set of data (Figure 5). This data had a logarithmic trend. This is because for concentric electrodes, the relationship between electric field and electric potential is 𝐸(𝑟) =

&4 , ./ 0 1./2(3) 5

(3)

where V0 is the power supply voltage (10V). Taking into account the error bars, the data points fit well in the logarithmic fit.

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Electric Potential V (V) vs. Average distance ravg (cm) 7 6.5

Electric Potential (V)

6 y = -3.255ln(x) + 9.2516 R² = 0.99863

5.5 5 4.5

Observed

4

Theoretical

3.5

y = -4.53ln(x) + 10.788 R² = 1

3 2.5 2 2

3

4

5

6

7

Average distance ravg (cm) Figure 5. Electric potential vs. average distance. Electric potential is on the y-axis; average distance is on the x-axis. Error bars on each point signify calculated error in both V and ravg. The logarithmic fit was represented by the equation y= -3.255ln(x) + 9.2516. R2= 0.99863. Set of data in blue represent observed data; set of data in red represent theoretical data.

The theoretical electric potential V was calculated using our data: 𝑉789:597; ∗

./ 0 1./2(5) ./ 0 1./2(3)

.

(4)

The theoretical electric potential at each average distance ravg was calculated and visualized with the observed data (Figure 5). Although the observed values follow a similar trend as the expected one, there are significant deviations. In addition to the large error in electric potential due to large variations caused by small changes in pressure, there were additional causes of error. Systematic error arose from the use of faulty equipment. The surface of the circular electrode was not uniform, thus the amount of contact between the electrode and the paper was likely inconsistent. To elaborate further, when there was likely good contact between the electrode and the paper in some spots, there was poor to no contact between the electrode and the paper in other spots. This intermittent contact likely substantially contributed to deviations from the expected electric potential. The faulty equipment contributed to the error in electric potential, further supporting that the estimated 𝛿V should have been larger.

Conclusion The relationship between electric field and electric potentials was examined in these experiments. Investigation 1 measured the electric field between parallel electrodes. Through measurements and calculations of electric potential, electric field lines, distance between adjacent points, and distance from

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the negative electrode, theoretical values of electric field and electric potential were calculated. The observed electric field was E= 0.954 ± 0.0202 V/cm (Figure 1). The theoretical electric field was 1.0 V/cm. Although these values are similar, they are not equal within calculated error. This indicates additional experimental error, as well as the likelihood that the error in electric potential was not fully accounted for. Investigation 2 measured the electric field between concentric electrodes. Similar measurements were recorded, and observed electric potential was compared to expected electric potential. These observed values deviated greatly from the theoretical values. In addition to error in electric potential that was not fully accounted for, the surface of the circular electrode was not uniform. This created additional deviation in results.

Questions 1. The electric field in between the parallel electrodes point from positive to negative, or right to left.

2. The electric field lines for the circular electrodes pointed from positive to negative, or away from the center.

3. The average electric field between the parallel electrodes was smaller in the fringe region than in the central region. As you approach the fringe region, the lines curve, increasing the distance and thus decreasing the electric field. 4. (Investigation 1) Assuming the power supply remains at 10 V, the magnitude of the electric field changes will double if the distance d between electrodes is halved. 𝑉 𝑥 = −𝐸𝑥 102𝑉 𝑥 = −𝐸(0.05𝑚) &

𝐸 = 200 F = 2.02𝑉/𝑐𝑚 When d=10cm, the magnitude of the electric field was 1.0 V/cm. When d= 5cm, the magnitude of the electric field becomes 2.0 V/cm.

5. (Investigation 2) If the diameter of the outer ring is doubled and everything else remains unchanged, the electric field (c) it changes but it neither doubles nor becomes half. This relationship is represented by the equation: 𝐸 𝑟 =2

4 & ./ 0 1./2(3) 5

,

where b is the diameter of the outer ring, then as b increases,

&./ 0 1./2(3)

decreases, and thus E

decreases. However, doubling b does not double ln(b), so E is not halved (nor doubled).

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6. Coulomb’s Law states that the electric field is equal to Coulomb’s constant, k, times two multiplied charges (Q1*Q 2), divided by the squared distance between the two charges, or 𝐸 =

IJKJK 5L

. Additionally,

to find electric field from Coulomb, the electric field times the distance is equal to k times charge Q divided by the distance. In other words, electric field E is proportional to 1/r2. In Investigation 1, the electric field E remained constant. This is because the power supply was set to 10V, and the distance between the two electrodes was 10cm, so the electric field at any point was 1V/cm. However, in Investigation 2, E was proportional to 1/r. If you have a region of charge, you need to integrate over that region to find the force due to all of the charge. Thus, the relationship between electric field and distance becomes E ∝1/r.

Acknowledgements I would like to thank my lab partners, PARTNER's NAME and ANOTHER, for our collaboration in order to successfully complete this experiment. Additionally, I appreciate the guidance given to us from our TA, Kai.

References [1] O.Batishchev and A.Hyde, Introductory Physics Laboratory, pp 161-170, Hayden-McNeil, 2016. [2] "Straight Line Fit – IPL." Introductory Physics Laboratory. 2016. http://www.northeastern.edu/ipl/data-analysis/straight-line-fit/.

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