Field Mapping - Lab Report

Abstract For this experiment, we had to identify the nature of electric fields set up by three different electrode arrangements via equipotential surfaces. The first arrangement was simply two electrodes in the conductive paper. The second arrangement was those same two electrodes in the same location, but conductive ink was drawn in a straight, vertical line through the electrode holes. The third arrangement was two electrode holes alone two conductive ink lines that made a cathode ray tube (CRT) shape. It was found that the further the probe was from the negative electrode, the higher the electric potential difference (voltage). The other trend observed was that as the equipotential voltage increased, the magnitude of the electric field increased. Goals of Experiment The goal of this experiment was to determine how different electrode arrangements affect the electric field. The electric field can be used to determine the electric potential at any point in the field. Introduction When a charged particle is placed in space, it exerts an electric field on all other regions of space. The electric field quantity is a vector, meaning it has both a magnitude and direction. The magnitude and direction of the electric field depends on the location and arrangement of the source charges. The field is dependent on what sort of charge this particle has. The electric field points towards negative charges and away from positive charges. When an electron is placed in an electric field, it will move from regions of low electric potential (more negative areas) to regions of high electric potential (more positive areas). This logic is opposite for a proton. The electric field can be determined by identifying areas with the same electric potential difference (voltage) around source charges. These regions with the same voltage are known as equipotential surfaces. If equipotential surfaces are found for a charged particle (or several charged particles), the electric field will be perpendicular to these surfaces. This is because there would be no force on a charge in the direction of the surface. Theory and Derivations The electric potential (V) is defined as the electric potential energy per unit of charge at that point. However, the electric potential at any given point in space is insignificant. What is important is the magnitude of the electric potential difference (∆V) between two points, especially two points in an electric field. The potential difference between these two points can be used to find the electric field using the equation E= ∆V d (1) where E is the magnitude of the electric field, ∆V is the electric potential difference between the two charges at that point, and d is the distance from the charge. Per this equation, the magnitude of the electric field is directly proportional to the electric potential difference and inversely proportional to the distance. Procedure Figure 1. The first electrode arrangement. Electrodes were placed in the holes on the conductive paper. Conductive ink was drawn in a circle around the holes. Figure 2. The second electrode arrangement. Electrodes were placed in the holes on the conductive paper. Conductive ink was drawn in a line passing through the holes. Figure 3. The third electrode arrangement. Electrodes were placed in the holes on the conductive paper. Conductive ink was drawn in a funnel shape passing through the holes. For this experiment, three different electrode arrangements were used. These three arrangements can be seen in Figures 1-3. The same procedure was carried out for each one. The conductive paper was placed on a mounting board corresponding to where the holes were (there were two different boards). Before any values were recorded, the wires were attached to the power supply and the conductive paper. The power supply was turned on and was adjusted to 10 volts. This ensured that the highest voltage recorded on the conductive paper would be 10 volts. The voltmeter was then attached to the negative electrode, and the probe, also attached to the voltmeter, was used to touch the conductive paper and record the voltage at the point it was touched at. Once similar voltages were found on the paper, they were recorded on the given graph paper. When five of the same voltages were found, a line was drawn connecting the same voltages. Five different voltages were found for each electrode arrangement. This gave us five different equipotentials in each field map. Finally, the electric field lines were added according to the equipotentials found. The positions on the conductive paper as well as the voltages associated with those positions were then used to calculate the magnitude of the electric field and the electric potential associated with that field. Sample Calculations and Results This lab required us to calculate the magnitude of the electric field at various equipotential surfaces. This is found by utilizing Equation 1 above. The electric potential difference (∆V) was determined using the voltmeter in the lab. The probe, when placed on the conductive paper, told us the potential difference between the two electrodes at that point. The distance used was the distance from the negative electrode. This distance was found by either counting the squares if it was in a straight line, or using the Pythagorean theorem if the distance between the charge and the equipotential point had x- and y-components. E= 1.7V V ∆V = =48.571 d 0.035m m (2) The rest of the calculations can be seen in Tables 2, 4, and 6. For each equipotential surface, only one point was used to calculate the electric field for that equipotential surface. The point I chose to use was the point in that equipotential surface with the lowest x-position according to the origin. If two points had the same x-positon, I used the one that had the lower y-position. I chose this for each one to keep the calculations as consistent as possible. Discussion and Conclusions The overall goal of the experiment was to determine the magnitude of the electric field of the different equipotential surfaces for each electrode arrangement. This goal was met because five different equipotential surfaces were found for each electrode configuration, and the voltages for these equipotential surfaces were used to calculate the various magnitudes of the electric fields. We attempted to understand the three-dimensional concept of equipotential surfaces using only a two-dimensional piece of conductive paper. In reality, these equipotential surfaces would span a three-dimensional area. As an example, we will look at the first arrangement, with just the two electrodes. The equipotential “lines” found appeared to just be circles surrounding each charge. However, these circles are actually spheres, encompassing a three-dimensional space. This lab only utilized a two-dimensional piece of paper, but it is important to realize that these equipotential surfaces expand in all directions in three-dimensional space. All of the positions and the electric potential differences, also known as voltages, at those positions are compiled in Tables 1, 3, and 5. There is one table for each arrangement. Next, the magnitudes of the electric fields were calculated using the voltages and the distances from the negative charge. These results can be seen in Tables 2, 4, and 6. The relationship between the distance, the voltage, and the electric field magnitude can be seen in Graphs 1, 2, and 3. Each graph represents the relationship for a different electrode arrangement. Two general trends were found after all the data was collected. The first trend was that as the distance from the negative electrode increased, the voltage increased. This is because the positive electrode is associated with high electric potential, as previously described. The second trend observed was that as the voltage increased, the magnitude of the electric field also increased. This result was surprising to me. In an ideal system, the electric field should be constant everywhere unless the source charges change (the actual charge changes, the source charge moves, etc.). The magnitude of the electric field is affected by the electric potential difference and the distance of that difference from the source charge. According to Equation 1, electric potential is directly proportional to the magnitude of the electric field, while the distance is inversely proportional to the magnitude of the electric field. Both the electric potential difference and the distance increased. Ideally, both of these values increasing should, in a way, cancel each other out, so the electric field is the same everywhere on the conductive paper. According to the data we obtained from this experiment, either the voltage increased too much, or the distance did not increase enough. This could be contributed to any of the errors described next. One source of error in our measurements could be the fact that the conductive paper was not accurate. This paper was used over the course of many labs, and it could have been damaged, giving us incorrect voltage readings. Another source of error could be that the probe was not placed exactly on the points we recorded. The voltmeter was very sensitive at some locations on the conductive paper, and the value would bounce around a lot. At some points, we decided to choose the value the voltmeter stayed on the longest, which may not have been accurate or the most effective way to collect some of the values. Finally, a source of error could be the points I decided to use for the electric field calculations. Each point could have given me a different magnitude for the electric field pertaining to that equipotential surface, and one point may have been more superior than another when doing the calculations, but I was simply choosing points to keep it as consistent as possible. First Electrode Arrangement Voltages X-Position (m) 0.02 0.03 0.06 0.06 0.075 0.01 0.02 0.04 0.04 0.08 0.14 0.14 0.14 0.14 0.14 0.18 0.22 0.22 0.24 0.27 0.20 0.22 0.22 0.24 0.255 Y-Position (m) 0.10 0.09 0.08 0.12 0.10 0.155 0.055 0.06 0.15 0.10 0.00 0.02 0.04 0.12 0.15 0.10 0.00 0.18 0.19 0.20 0.10 0.08 0.12 0.09 0.10 Voltage (V) 1.7 2.2 5.1 6.7 8.3 Table 1. Voltage readings at various x- and y-positions on the conductive paper for the first electrode configuration. First Electrode Arrangement Electric Field Distance from Negative Electrode (m) Voltage (V) 0.035 1.7 0.071 2.2 0.131 5.1 0.125 6.7 0.145 8.3 Electric Field (V/m) 48.571 30.986 38.931 53.600 57.241 Table 2. Magnitude of the electric field for the first electrode configuration calculated from voltage and distance. Second Electrode Arrangement Voltages X-Position (m) 0.09 0.09 0.09 0.09 0.09 0.11 0.11 0.11 0.11 0.11 0.13 0.13 0.13 0.13 0.13 0.15 0.15 0.15 0.15 0.15 0.17 0.17 0.17 0.17 0.17 Y-Position (m) 0.01 0.04 0.08 0.12 0.16 0.00 0.03 0.07 0.12 0.15 0.02 0.06 0.10 0.12 0.16 0.00 0.04 0.08 0.12 0.16 0.00 0.02 0.06 0.10 0.16 Voltage (V) 1.6 3.0 4.3 5.7 7.6 Table 3. Voltage readings at various x- and y-positions on the conductive paper for the second electrode configuration. Second Electrode Arrangement Electric Field Distance from Negative Electrode (m) Voltage (V) Electric Field (V/m) 0.095 1.6 16.842 0.112 3.0 26.786 0.106 4.3 40.566 0.135 5.7 42.222 0.149 7.6 51.007 Table 4. Magnitude of the electric field for the second electrode configuration calculated from voltage and distance. Third Electrode Arrangement X-Position (m) 0.16 0.18 0.19 0.20 0.21 0.07 0.09 0.12 0.13 0.14 0.16 0.17 0.19 0.20 0.22 0.05 0.08 0.09 0.13 0.15 0.07 0.10 0.12 0.13 0.14 Y-Position (m) 0.90 0.80 0.0775 0.0725 0.07 0.10 0.10 0.10 0.10 0.10 0.1025 0.10 0.095 0.0925 0.09 0.11 0.11 0.11 0.11 0.11 0.12 0.12 0.12 0.12 0.12 Voltage (V) 0.7 2.4 3.0 5.3 7.9 Table 5. Voltage readings at various x- and y-positions on the conductive paper for the third electrode configuration. Third Electrode Arrangement Electric Field Distance from Negative Electrode (m) Voltage (V) 0.050 0.7 0.041 2.4 0.052 3.0 0.063 5.3 0.050 7.9 Electric Field (V/m) 14.000 58.537 57.692 84.1270 158.000 Table 6. Magnitude of the electric field for the third electrode configuration calculated from voltage and distance. First Electrode Arrangement Electric Field 9 70 8 60 50 Voltage (V) 6 5 40 4 30 3 20 Electric Field (V/m) 7 Voltage Electric Field 2 10 1 0 0.04 0.07 0.13 0.13 0 0.15 Distance from Charge (m) Graph 1. Voltage readings and electric field magnitudes for five select points for the first electrode arrangement. As the voltage increases, the magnitude of the electric field increases. Second Electrode Arrangement Electric Field 8 60 7 50 40 5 4 30 3 20 2 Voltage Electric Field 10 1 0 0.1 Electric Field (V/m) Voltage (V) 6 0.11 0.11 0.14 0 0.15 Distance from Charge (m) Graph 2. Voltage readings and electric field magnitudes for five select points for the second electrode arrangement. As the voltage increases, the magnitude of the electric field increases. 180 8 160 7 140 6 120 5 100 4 80 3 60 2 40 1 20 0 0.05 0.04 0.05 0.06 Electric Field (V/m) Voltage (V) Third Electrode Arrangement Electric Field 9 Voltage Electric Field 0 0.05 Distance from Charge (m) Graph 3. Voltage readings and electric field magnitudes for five select points for the third electrode arrangement. As the voltage increases, the magnitude of the electric field increases.