Electric Field and Electric Potential Lab Report

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Feb 20, 2024

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Report for Experiment #16 Electric Field and Electric Potential Vaughn Montoya Lab Partner: Sen Niu TA: Ji Tae 2/17/2023

Introduction An electric field happens to be a vector force that encompasses an charged related to the forces between them. There happens to be one way to detect it which is to see if another charge experiences an repulsion or attraction in the field. Electric fields can be seen by electric field lines which point in the direction of the maximum field at that point. Electric field lines are at every point perpendicular to the equipotential lines. The strength of a force on a test charge is F=qE (1) where q is the test charge and E is the electric field. The potential field is a scalar field, which is giving just magnitude and the electric field is a vector field, which has both magnitude and direction. When connecting pieces of metal to two poles of a battery or DC power supply positive charges will accumulate on one of the pieces while the other will accumulate negative charges. These charges will make a potential field between the two pieces of metal. Then a ground, zero potential was set to one of the electrodes and compared the other points thereby making a map of potentials. A simple way to do this is to look for lines along the potential that stay the same which are known as equipotential lines. Electric field can be calculated by using the following equation: ࠵? ! = − ∆࠵? ∆࠵? = − ࠵? " − ࠵? # ࠵? " − ࠵? # (2) The objects of this lab were to look at the electric potential between two electrodes, and to figure out the relation between electric potential and electric field as well as to study electric field between electrodes. In the first Investigation a piece of black conducting paper was placed in the middle with parallel electrodes and lines drawn long the edges of the electrodes. They were then connected to a voltage source and equipotential lines were searched for between the electrodes. In this Investigation voltage happens to increase from negative to positive electrode and the electric field between the parallel plates remains approximately constant. A plot of V vs. X and E vs. x was made to visualize this theory. Investigation 2 used the same black conducting paper was used and two brass circular electrodes were used this time. In this Investigation static field in a coaxial cable, like those used to feed cable TV was stimulated to highlight that equipotential lines are curved. Plots of V vs. average radius and E vs. 1/࠵? $ were made to look at these theories. Investigation 1 The setup for Investigation 1 consisted of placing the rubber pad in the middle of the table and then placing the conducting paper on the top. Then two parallel plates were placed 10 cm apart from each other. A grease pencil was used to trace a line on the inside of the electrodes to highlight where they were. 10 V is what the power supply was set to and the connecting wires which were put in and attached and the plate on the left was set to x=0 also known as the grounding plate while the one on the right was set x=d. The scale of the voltmeter was set to read at 10 V. The probe was used to check that the positive plate, the plate on the right registered 10 V. Then equipotential lines were found in increments of 1V, the 5 V line was found first because towards the center should be parallel, a straight line. Then 2 V to 8 V was found using the probe and marked and labeled with a grease pencil. Also, the potential was found outside of the two parallel plates

known as the fringe field, where the electrode ends. One person was holding down on the parallel plates to ensure there was a solid connection while the other was doing the measurements. The 5v was then confirmed to be halfway between the two electrodes. Then perpendicular lines were drawn from the equipotential lines. Measurements between the equipotential line and grounded electrode were recorded. Half the thickness of line was determined to be the uncertainty of x which was simply just 0.0005m. Error in voltage was just 1% of the voltage value. The theoretical voltage V was found by using the following equation: ࠵? = |Ex | (3) of the Then Electric field, E was calculated between pairs of potential lines and its error was found using the following equation: ࠵?࠵? = ࠵? 4 ࠵?࠵? % ࠵? + ࠵?࠵? % ࠵? (4) Then the theoretical electric field was calculated by taking the absolute value of second equation in the introduction. Then the average position, ࠵? #&’ was found using the following equation: ࠵? #&’ = ࠵? # + ࠵? " 2 (5) Then its error was found using the following using the following equation: ࠵?࠵? #&’ = 1 2 8 2࠵?࠵? % (6) A table was made for the data in Investigation 1 Table 1: All data from Investigation 1: Vth= Theoretical Voltage and Eth= Theoretical Electric Field x (m) δx (m) V (V) δV (V) ࠵? #&’ (m) ࠵?࠵? #&’ (m) Ε (V/m) δE( V/m) Vth (V) Eth( V/m) 0.018 0.0005 2 0.02 0.032 0.0035 -100 -3.7 -1.8 -100 0.028 0.0005 3 0.03 0.047 0.0035 -100 -5.1 -2.8 -100 0.038 0.0005 4 0.04 0.064 0.0035 -71.4 -4.9 -3.8 -100 0.052 0.0005 5 0.05 0.083 0.0035 -111 -9.8 -5.2 -100 0.061 0.0005 6 0.06 0.098 0.0035 -83.3 -8.8 -6.1 -100 0.073 0.0005 7 0.07 0.11 0.0035 -125 -15.5 -7.3 -100

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0.081 0.0005 8 0.08 x 0.0035 -98.8 x -8.1 -100 A plot of Voltage (V) vs position (X) was made to look at voltage as it increases from negative to positive electrode. Plugging the data into the IPL straight line of fit calculator, the experimental slope was found to be 92.0 ࠵?/࠵? ± 1.19 ࠵?/࠵? . The experimental slope did not fall within the theoretical slope. This could have been because of human error in measuring and drawing the equipotential lines. Figure 1: Voltage (V) vs. Position (X) A plot of E vs. X was made to look at and show that electric field remains constant between the two electrodes. Using IPL straight line of fit calculator, the experimental slope was found to be 165.0 ࠵?/࠵? % ± 107.2 ࠵?/࠵? % . The slope does not agree with the theoretical data which could be because of human errors such as not correctly measuring the distance between grounded potential and equipotential lines thus impacting electric field calculations. y = 92.444x + 0.3646 y = 100x 0 1 2 3 4 5 6 7 8 9 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Voltage (V) Distance (m) Voltage (V) vs Distance(X) and V Theoretical Expected Theoretical Linear (Expected) Linear (Theoretical) Linear (Theoretical)

Figure 2: Electric Field (E) vs. Position (X) Investigation 2 The setup for Investigation 2 consisted of using the same black paper underneath the soft rubber pad. Two circular brass electrodes were used in this Investigation. The bigger one put in the center and the and traced with a grease pencil as well as an x-y coordinate system was set up in the inside of this. The small brass ring was placed in the center of the bigger ring. Then we measured the radius and distances between origins and equipotential lines. The radius of the center electrode a was measured to be 0.01 ± 0.0005 ࠵? and the outer electrode was measured to be 0.1075 ± 0.0005 ࠵? . The error of these measurements was simply just half the smallest increment of the ruler. The outer ring was connected to grounded negative terminal while the inner one was connected to the positive terminal. Just as Investigation 1 the power supply and voltage were set 10 V. The prob-tipped wire was connected to ensure that the outer ring was 10 V. Then the equipotential lines were found starting at 5 V then 2 V and up to 7 V using the probe and marked at every 45° . Just as in Investigation 1, one partner did the measurements while the other put pressure down on the ring to ensure solid contact everywhere. Measured points were connected to the equipotential lines which happen to be roughly circular. The ࠵? ( distance was found from the coordinate origan to the equipotential lines, for each coordinate axis to get four distance measurements: r1 to r4. The average radial distance ࠵? #&’ was found using the following equation: ࠵? #&’ = ࠵? ) + ࠵? % + ࠵? * + ࠵? + 4 (7) The error in the average value δr for each equipotential lines using the following equation: y = - 100 -160 -140 -120 -100 -80 -60 -40 -20 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Electric Field E (V/m) Position x (m) Electric Field vs Average x Experimental Theoretical Linear (Theoretical)

࠵?࠵? #&’ = 1 √4 4 (࠵? ) − ࠵?) % + (࠵? % − ࠵?) % + (࠵? * − ࠵?) % + (࠵? + − ࠵?) % 4 (8) Then the Theoretical Voltage (V) was found using the following equation: ࠵? ,- = ln E ࠵? ࠵? G ln E ࠵? ࠵? G ࠵? . (9) Then the Theoretical Electric field was calculated using the following formula: ࠵?(࠵?) = ࠵? . ln ( ࠵? ࠵? ) 1 ࠵? (10) Table 1: Data from Investigation 2 ࠵? ) (࠵?) ࠵? % (࠵?) ࠵? * (࠵?) ࠵? + (࠵?) ࠵? #&’ (࠵?) ࠵?࠵? #&’ (࠵?) V (V) δV (V) 0.074 0.077 0.092 0.076 0.080 0.00025 2 0.02 0.053 0.073 0.066 0.058 0.063 0.00025 3 0.03 0.044 0.06 0.051 0.048 0.051 0.00025 4 0.04 0.034 0.041 0.045 0.038 0.040 0.00025 5 0.05 0.031 0.032 0.029 0.025 0.029 0.00025 6 0.06 0.02 0.023 0.026 0.02 0.022 0.00025 7 0.07 Table 2 : More data for Investigation 2 ࠵? $ ࠵?࠵? 0 1/࠵? $ (1/m) ࠵?( ) 1 ! ) (1/m) E(V/m) ࠵?࠵? (V/m) Theoretical Voltage (V) Theoretical E (V/m) 0.071 0.050 14.1 0.052 59.2 4.72 1.26 80.8 0.057 0.040 17.7 0.071 74.4 4.65 2.28 103.1

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0.045 0.032 22.2 0.067 93.3 4.74 3.16 127.0 0.034 0.024 29.1 0.059 122.5 4.84 4.22 163.1 0.026 0.018 38.8 0.053 163.5 4.78 5.48 220.3 0.011 0.0079 89.9 0.065 378.5 8.42 6.63 289.6 A plot of V vs. average radius ( ࠵? #&’ ) was made using the logarithmic line compared to the linear one because equipotential lines are curved due to the circular brass electrode. The experimental and theoretical values do not each other and do not fall within uncertainty because the errors bars do not encompass each other. Figure 3: Voltage (V) vs. average radius (ravg) The electric field was then calculated using equation (9). The location rE is approximately halfway between the equipotential lines. The values of 1/re were found by calculating the midpoint of the equipotential lines and dividing by 1. Then a plot of E vs. 1/re was made. The experimental slope calculated using IPL straight line of fit calculator, which was found to be, 4.21 ࠵? ± 0.116 ࠵? These values do not encompass the uncertainty of the theoretical line. 0 1 2 3 4 5 6 7 8 0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 Voltage (V) average radius (m) Voltage vs Average Radius Experimental Theoretical Log. (Experimental) Log. (Theoretical )

Figure 4: Electric Field E vs 1/rE Conclusion The relationship between electric field and electric potential was highlighted by parallel electrodes placed 10cm apart. The left one was the grounded one, x=0 while the one on the right was set to x=d. The probe was used to locate the equipotential lines between 2 V and 8 V. The 5 V line was expected to be right in the middle and parallel because it was directly in the center of the field. The distance between the grounded electrode and the equipotential lines was measured. Then the average position was calculated. Then a plot of V vs. x and E vs. x was made to highlight that voltage increases the closer it gets to the positive terminal and that the electric field between two electrodes remains constant. The experimental slope for V vs. X was found to be 92.0 ࠵?/࠵? ± 1.19 ࠵?/࠵? from IPL straight line of fit calculator and the theoretical was mapped on the graph. The experimental slope was not within uncertainty which could have been because of human error in measuring the equipotential lines. Using IPL again the slope of the E vs. x graph was found to be 165.0 ࠵?/࠵? % ± 107.2 ࠵?/࠵? % which does not fall within uncertainty. This could have been because of errors is measuring which effected the E field calculations. Some potential improvements could have been using a thinner pencil to decrease the error in distance measurement or simply just taking longer to precisely measure each line. In Investigation 2, a different part of the black conducting paper was used, and two circular brass electrodes were used. An x-y coordinate system was made and the radius of the center electrode a was measured to be 0.01 ± 0.0005 ࠵? and the outer electrode b, was measured to be 0.1075 ± 0.0005 ࠵? . A plot of V vs. Average radius was made with a logarithmic tread line to highlight that equipotential lines are curved because of the circular brass electrode. The experimental slope was found, and it was not within uncertainty because the error bars don’t encompass the theoretical tread line. Then a graph of E vs. 1/re was made and the experimental slope was found to be, 4.21 ࠵? ± 0.116 ࠵? . These values are within 0 50 100 150 200 250 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 E-Field (V/m) 1/rE (1/m) Electric Field vs 1/electric field radius Experimental Theoretical Linear (Experimental ) Linear (Theoretical)

uncertainty. A thinner grease pencil would have been better to decrease error in measurements and better draw the equipotential lines. Questions 1. Since potential difference and electric field are proportional to one another the appearance of lines would stay the same. 2. This would happen to produce a systematic error. 3. The average electric field in the fringe region would be smaller than the central because the equipotential lines become greater the further the probe moves from the center electrode. 4. If the power supply stays 10 V while the distance d is halved the magnitude of the electric field would double because distance and electric field happen to be inversely proportional. 5. c.) it changes but it neither doubles nor becomes half This is true since in Investigation 2 the equation is more complex than that of the one used in the first Investigation.

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