Experiment 16 - Declan Rogers

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Northeastern University *

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Electrical Engineering

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Jan 9, 2024

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Report for Experiment #16 Electric Field and Electric Potetial Declan Rogers Lab Partner: Aiden Kaneshiro TA: Zhuyao Wang 5-26-22 Abstract This experiment studied the electric potential and the electric field created between two parallel electrodes and two concentric electrodes. The first investigation studied the parallel electrodes and was intended to determine the electric potential through experimentation. This was found to be 0.9785 V/cm with an allowance of 0.00464 which was consistent with the expected theoretical value of 1 V/cm. The electric field was then determined to be -0.0058 V/cm^2 with an allowance of 0.00752 which falls near the expectation of 0 V/cm^2. This value was found by determining the location of the equipotential lines for 2 V through 8 V and then the distance between these lines was analyzed. The second investigation studied the concentric electrodes and was intended to determine the electric potential and electric field. To find these values the equipotential lines were found at 45-degree intervals from 2 V to 7 V between the two electrodes. The distances between these lines were then used to determine the magnitude of electric potential and the electric field which both agreed with their theoretical values.

Introduction This experiment studies electric fields and electric potential through two separate investigations. Electric fields are the fields created due to interaction between charges and these fields can create forces. The first investigation of this experiment utilized two charged rods to create an electric field on a piece of conductive paper to measure the voltage or electric potential at different places on the paper. The voltage was measured using a voltmeter and pressing the leads to different places on the paper between the charged rods. In similar fashion, the second investigation utilized the voltmeter to gather data but instead between two circular charges. The intention of this experiment was to locate were the voltage reaches specific amounts and utilize that data to calculate the electric field. In the first investigation, the electric field is expected to be constant. As for the second investigation, the electric field should increase in a linear fashion, with the voltage increasing with the distance proportionally. Graphs were then used to visualize the difference in behavior across the investigations and ensure that they are consistent with the expected outcome. Investigation 1 To set up the first investigation, place the rod electrodes on the conductive paper. Ensure that you have enough space on the paper to include the second investigation without overlap. Ensure that the two electrodes are 10 centimeters apart and connect them to the power supply such that the negative lead is connected to the left electrode. Place the voltmeter lead on the negative end of the electrode setup. Modify the voltmeter settings to ensure it reads to a precision maximum of 20 volts. Set the power supply to 10 volts and tweak the voltage until the voltmeter reads 10 volts exactly. Once the setup is complete, drag the voltmeter from the negative side until it reads 2 volts and mark this place with a grease pencil. Repeat this action for every 1-volt interval until you get to 8 volts, writing the voltage value at each equipotential line. Utilize the voltmeter to then draw the equipotential lines at each value past the extent of the electrodes into the fringe. Recognize that the equipotential lines will curve as you leave the space directly between the electrodes, slowly spacing out. Then, with the grease pencil, designate the direction of the electric field on the paper.

Table 1: Equipotential Line Distance from Grounded Electrode Voltage (V) δV (V) δV/V (V) Distance from electrode (m) δx (m) δx/x (m) 2 0.02 0.01 0.019 0.00025 0.00013 3 0.03 0.01 0.030 0.00025 0.00008 4 0.04 0.01 0.040 0.00025 0.00006 5 0.05 0.01 0.051 0.00025 0.00005 6 0.06 0.01 0.060 0.00025 0.00004 7 0.07 0.01 0.070 0.00025 0.00003 8 0.08 0.01 0.081 0.00025 0.00003 Absolute error was found through the defined error of uncertainty in the voltage, 1%, by dividing error by the measured voltage value The uncertainty of the position values for the equipotential lines was found through determining half of the thickness of the grease pencil line, 2.5 mm. The absolute error was then found by dividing the error by the position value. With the distance between each line and the voltage values, the equation below was used to determine the E-Field for each set of lines: Equation 1: E ( x ) = V a − V b x a − x b The error in the electric field was then found using the equation Equation 2: δE = √ ( δ ∆V ∆V ) 2 + ( δ ∆ x ∆ x ) 2 Both δ ∆x and δ ∆V are needed to determine δE , therefore the following two equations are used to determine those values:

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Equation 3: δ ∆x = √ ( δ x b ) 2 +( δ x a ) 2 Equation 4: δ ∆V = √ ( δ V b ) 2 +( δ V a ) 2 The average position between the two equipotential lines is then found using the equation: Equation 5: x avg = x a + x b 2 Then the error in the average is found with the following: Equation 6: δ x avg = √ δ x a + δ x b Table 2: Electric field determined using Equation 1 at the average position between two equipotential lines Voltage (V) δ ∆ V (V) δ ∆ V/ ∆ V (V) ∆ x (m) δ ∆ x (m) δ ∆ x/ ∆ x (m) 2-3 0.0360 0.0141 0.011 0.00354 0.00321 3-4 0.0500 0.0141 0.010 0.00354 0.00354 4-5 0.0640 0.0141 0.011 0.00354 0.00321 5-6 0.0781 0.0141 0.009 0.00354 0.00393 6-7 0.0921 0.0141 0.010 0.00354 0.00354 7-8 0.1060 0.0141 0.011 0.00354 0.00321 Electric field (V/cm) δE (V/cm) Average position (m) δ x avg

(m) -0.909 -0.292 0.0245 0.00177 -1 -0.354 0.0350 0.00177 -0.909 -0.292 0.0455 0.00177 -1.11 -0.437 0.0555 0.00177 -1 -0.354 0.0650 0.00177 -0.909 -0.292 0.0755 0.00177 Graph 1: Voltage as a function of the distance from the grounded electrode With the IPL line fit calculator, the slope was found to be 0.9785 V/cm with an uncertainty of 0.00464 V/cm. The theoretical slope of this graph was 1 V/cm, with each step in 1 volt equivalent to a step in 1 centimeter. As the distance of the electrode increased, the voltage at that line would increase proportionally with a factor of 1. The theoretical slope supports the data gathered and the calculated slope value of 0.9785 V/cm. voltage (v) vs distance (cm) y = 0.9785x + 0.0935 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 distance from electrode (cm)

Graph 2: The electric field as a function of average distance from the grounded electrode between two equipotential lines. With the IPL line fit calculator, the slope was found to be -0.0058 V/cm^2 with an uncertainty of 0.00752 V/cm^2. The theoretical value of the slope of electric field versus distance was 0, with a constant electric field of -1. That means that the electric field does not change regardless of the location the electric field would be read as -1 N/C. The calculated value for the slope is within a degree of uncertainty from 0 and is therefore consistent with the expected value. Therefore, the investigation was successful in representing the behavior of two infinite charges. Investigation 2 The second investigation is set up starting with the two circular electrodes: the ring and the cylinder. Place them in the leftover space on the conductive paper and center them on two perpendicular lines drawn with the grease pencil. Measure the diameter of each electrode, ensuring to measure the electrode portion of the cylinder as well. Connect the grounded wire of the power supply to the outer ring and connect the positive wire to the inner cylinder. Connect the grounded lead of the voltmeter to the outer ring and the other lead to the positive cylinder. Set the power to 10 V and toggle the power until it reads 10V on the voltmeter. Once the setup has been complete, begin by placing the probe end of the voltmeter on the conductive paper where it reads 5 volts and mark this point at 45-degree intervals around the entire circle. Repeat that process for each voltage value from 2 volts to 7 volts. Draw equipotential circles that intersect each mark to delineate the equipotential zones. Following all of the circles, disconnect the power supply. Electric Field (V/cm) vs. avg. distance (cm) y = -0.0058x - 0.9439 0 0 1 2 3 4 5 6 7 8 -0.2 -0.4 -0.6 -0.8 -1 -1.2 Average Distance (cm)

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With the equipotential lines, the radial distance is measured from the origin of the coordinate plane to the intersection of each equipotential line, resulting in 4 data points for each line. With these values the average radius was found with the following equation: Equation 7: r avg = r 1 + r 2 + r 3 + r 4 4 The error in the radius is then found using the following: Equation 8: δr avg = STD DEV r 1 + r 2 + r 3 + r 4 √ n − 1 Table 3: Radial distance at each equipotential line V (V) δV (V) δV/V (V) r1 (m) r2 (m) R3 (m) r4 (m) r average (m) δr average (m) 2 0.02 0.01 0.066 0.070 0.065 0.066 0.0145 0.000645 3 0.03 0.01 0.051 0.053 0.051 0.053 0.0213 0.000629 4 0.04 0.01 0.041 0.042 0.040 0.043 0.0300 0.000408 5 0.05 0.01 0.032 0.031 0.031 0.034 0.0383 0.000946 6 0.06 0.01 0.025 0.025 0.024 0.026 0.0488 0.000479 7 0.07 0.01 0.020 0.021 0.020 0.021 0.0603 0.000854 These values were used to graph V as a function of the average radius with a logarithmic fit to calculate the line of best fit of the recorded values. This is used because electric potential and radial distance are exponentially related with one another because the radius changes with respect to the natural log. The logarithmic fit is consistent with the data as graphed below, displaying that the electric field decreases with the radius. With the multiple values of the radius, the theoretical V can be found with the following equation: Equation 9: V theoretical = V O ln ( b ) − ln ( r i ) ln ( b ) − ln ( a )

Theoretical radius (m) Theoretical Voltage (V) 0.037 4.5012 0.046 3.5870 0.055 2.8360 These values are also found in the graph in orange: Graph 3: Voltage as a function of average radius from the center electrode Due to the proximity of the experimental values to the logarithmic best fit, the experiment is consistent with the expected theoretical results. With the examination of the electric potential, the electric field was examined similarly, using the same radii as earlier with the following equations: Equation 10: Voltage (v) vs Average Radius (cm) 8 y = 11.878e- 0.266x 7 6 5 4 3 2 1 0 0 1 2 3 4 Avg Radius (cm) 5 6 7 8 actual theoretical Expon. (actual)

∆r = r a − r b Equation 11: δ ∆r = √ ( δ r a ) 2 +( δ r b ) 2 Equation 12: r E = r a + r b 2 Then the following equation was used: Equation 13: E theoretical = ( V O ln ( b ) − ln ( a ) ) ( 1 r E ) To calculate the electric field for the experimental Voltage (V) δ ∆ V (V) δ ∆ V/ ∆ V (V) ∆ r (m) δ ∆ r (m) δ ∆ x/ ∆ x (m) 1/rE (1/m) Calculated E (V/m) 2-3 0.0361 0.0141 0.01475 0.00144 0.0000976 16.8 0.677 3-4 0.0500 0.0141 0.01050 0.00100 0.0000952 21.3 0.952 4-5 0.0640 0.0141 0.00950 0.00111 0.0001160 27.2 1.050 5-6 0.0781 0.0141 0.00700 0.00094 0.0001340 35.0 1.430 6-7 0.0922 0.0141 0.00450 0.00058 0.0001280 43.9 2.220 Theoretical 1/radius (cm^-1) Theoretical electric field (V/m) 20.0 8.410 30.0 12.610 40.0 16.810

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Graph 4: The graph of the electric field as a function of the reciprocal of r E with theoretical values displayed in blue. The experimental values of E determined and graphed are consistent with the theoretical values graphed alongside them, following lines of best fit with similar slopes. The theoretical electric fields at 1/rE 0.213, 0.272, and 0.439 cm -1 are 0.952, 1.050, and 2.220 V/cm which correspond with the experimental values of 0.841, 1.261, and 1.681 V/cm at 1/rE of 0.2, 0.3, and 0.4 cm. Conclusion This experiment examined and analyzed the electric potential and electric fields produced by parallel electrodes and concentric electrodes to understand their behavior in the real world, informed by their theoretical behavior. The parallel electrodes represented infinitely long electrodes and the concentric electrodes represented infinite cylinders. The first investigation determined the electric potential and field of the parallel electrodes to support that the electric field would be constant, and the electric potential would equal the product of the electric field and the distance. To do so, the equipotential lines for voltages 2 through 8 were gathered and the distance between them was recorded and analyzed to find the electric field. The experimental electric potential was determined to be 0.9785 V/cm with an allowance of 0.00464 which was quite close to the projected straight line with a slope of 1 found theoretically. The experimental electric field was determined to be -0.0058 V/cm^2 with an allowance of 0.00752 which proved to be extremely close to a slope of zero, supporting the theory that the electric field is constant. The second investigation was intended to find the electric potential and electric field in the concentric electrode setup. These calculated values would then be compared to the expected Electric Field (V/cm) vs 1 / Radius (cm- 1) 2.5 y = 5.368x - 0.2845 2 1.5 y = 4.2x + 0.001 1 0.5 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 1 / radius (cm)^-1 actual theoretical Linear (actual) Linear (actual) Linear (theoretical)

theoretical performance of the model to discern if the experiment properly represented the theoretical model of two infinite cylinders. In order to find the values, equipotential lines were found at 45-degree increments between the two circular electrodes from 2 volts to 7 volts. The radial distances were then gathered and analyzed to produce the electric field of the system. The electric field values were found to be 0.952, 1.05, and 2.22 V/cm at 1/rE of 0.213, 0.272, and 0.439 cm^-1, correlating to the theoretical values of 0.841, 1.261, and 1.681 V/cm at 0.2, 0.3, and 0.4 cm. These values lined up well in the graph and supported the expected theoretical output of the model. In order to improve the precision and accuracy of this experiment, more precise measuring and marking tools would be necessary to ensure that the measurements gathered are as close to absolute as possible. Added trials could also prove valuable to hone in on the most accurate values for each measurement. Questions 1. The equipotential lines would be at the same location but double the magnitude as they directly depend on the voltage. They would seem closer together because the distance between the plates would not change while the potential difference has increased. 2. It is systemic error as it can be assumed that the inconsistencies were evenly distributed and “consistent” among the data. It would be random error if the error actively skewed in one direction. 3. With the greater space between equipotential lines in fringe area, the electric field must be smaller in that space. 4. As in the first question, when the distance is halved but the electric potential is constant, the electric field must double. 5. The electric field will change however it will not be in a linear fashion as the electric field follows a logarithmic graph so the relationship will be similar.