RaghuChinta_Lab2_Reflection

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Apr 3, 2024

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Overview The purpose of this laboratory experiment is to explore and test the predictions of Gauss's Law and understand the relationship between electric potential and electric field. The focus is on a specific example involving parallel electrodes and how the electric field and potential vary in a two- dimensional system. Gauss's Law states that the total electric flux through a closed surface is proportional to the total charge enclosed by that surface. The equation is given by: E is the electric field, S is the closed surface, d A is a differential area element, Q enclosed is the total enclosed charge, and ε 0 is the permittivity of free space. Electric potential (V) is defined as the potential energy per unit charge at some point in space due to the presence of an electric field. The equation for electric potential is given by: The electric field (E) is related to the electric potential by the derivative: The specific example involves two parallel electrodes with uniform charge density. The electric field due to the negative electrode (E - ), positive electrod (E + ), and the total electric field (E) are calculated. The electric potential (V) between the electrodes is then derived, resulting in a linear relationship with distance (r).

There is an expected linear relationship between electric potential and distance. Experimental Setup, Data Analysis, and Plots Fig1a: 1-Dimensional Gauss’s Law Set up two parallel aluminum rods as electrodes, separated by about 10 cm. Connect the power supply to the electrodes to create a sheet-like "source of uniform charge." Measure potential differences along a line perpendicular to the electrodes, connecting their centers, using standard graph paper. Materials: 1. Tray of Shallow Tap-Water:

Purpose: The medium in which the behavior of the electric field and electric potential in two dimensions is explored. 2. DC Power Supply: Purpose: Supplies a potential difference (V PS ) between 2 electrodes, creating electric field in water. 3. Electrodes (Aluminum Rods and Ring): Purpose: Used to construct electric fields with different behaviors in the water. 4. Banana Wires: Purpose: Connect the power supply to the electrodes. 5. Digital Multimeter (DMM): Purpose: Measures potential differences (DC volts) between the ground electrode and selected points in the water. 6. Needle Probe: Purpose: Connected to the DMM to measure potential differences. Held vertically for precise measurements. 7. Graph Paper (Standard and Polar): Purpose: Used for measuring distances between points in the electric field. Standard graph paper for the parallel plates experiment, and polar graph paper for the point source and ring experiment.

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Fig1b: Point Source and Ring: 2-Dimensional Gauss’s Law Similar setup as Fig1a but with a radially symmetric electrode configuration with a rod as the positive electrode and a 6” diameter aluminum ring as the negative electrode instead of two parallel plates. Position the rod in the water to provide a point-like "source of charge," and center the aluminum ring on the positive electrode. Measure distances using polar graph paper due to the symmetry of the configuration. To collect measurements, we tested various points on the graph paper with the Needle Probe between the electrodes (Fig1a) and between the center and inner edge of the ring (Fig1b). Data Analysis & Plots

0 2 4 6 8 10 12 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 f(x) = 0.29 x + 1.38 R² = 0.99 Parallel Plate Electric Potential Distance (cm) Electric Potential (V) Fig2a : Variation in electric potential between two parallel plates at different distances. 0 1 2 3 4 5 6 7 8 9 10 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 f(x) = − 1.13 ln(x) + 3.84 R² = 1 Ring of Charge Electric Potential Distance (cm) Electric Potential (V) Fig2b : Radial Electric Potential profile around a charged ring at varying distances from the center.

0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3 3.5 4 f(x) = − 1.13 x + 3.84 R² = 1 Linearized Ring of Charge Electric Potential Ln(Distance (cm)) (unitless) Electric Potential (V) Fig2c : Logarithmic Radial Electric Potential profile around a charged ring. The x-axis is scaled logarithmically to provide a linear fit to the data. Result(s) and Comparison The experimental data for the Parallel Plates (Fig2a), Ring of Charge (Fig2b), and Logarithmic Ring (Fig2c) plots were fitted with equations: Parallel Plates: The fitted equation y = .2945x + 1.3848 indicates a positive linear relationship between the electric potential (y) and distance (x). This aligns with the expectation in the context of parallel plates, where a uniform electric field should result in a linear increase in electric potential with distance. The high R² value (0.9937) indicates a strong fit, supporting the validity of the model. Ring of Charge: The fitted equation y = -1.134ln(x) + 3.8383 suggests a logarithmic relationship between electric potential (y) and distance (x). This is in line with theoretical expectations, as the electric potential around a charged ring typically follows a logarithmic pattern. The high R² value (0.9995) further supports the accuracy of the logarithmic model. Logarithmic Ring: Interestingly, the Logarithmic Ring plot resulted in a fitted equation y = - 1.1336x + 3.8383, which closely resembles the equation for the Ring of Charge. This may be indicative of the experimental setup producing results that are more effectively modeled by a linear function when using a

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logarithmic scale for the distance axis. The high R² value (0.9995) suggests a strong correlation with the linear fit. Dominant Source of Uncertainty A dominant source of uncertainty is the human error in probe placement. The needle probe must be accurately placed at the measurement points. Human error in probe placement can introduce systematic errors, especially when measuring along specific lines (e.g., along electric field lines). The vertical orientation of the needle probe is crucial, and a slight deviation can affect the measurements. This led to errors in the measured values for the distances that contributed to the uncertainty in the following way especially for the δln(r) term of the logarithmic fit for the ring data where r is the radial distance from the center of the ring. To propagate this we used the formula: δln(r) = δr/r to get the error bars on the graph in Fig2c, which indicate significant fluctuations in the margins of error for any given distance. Conclusion The obtained results align well with the initial hypotheses based on the theoretical understanding of electric potential. The linear relationship for parallel plates and the logarithmic relationship for the ring of charge are consistent with expectations. The similarity between the fits for the Ring of Charge and Logarithmic Ring suggests that the choice of a logarithmic scale for the distance axis may impact the apparent functional form of the relationship. In conclusion, the experimental results support the theoretical predictions for electric potential in the given configurations, and the high R² values indicate robust fits to the data. The unexpected similarity in the fits for the Ring of Charge and Logarithmic Ring experiments warrants further investigation into the influence of the logarithmic scale on the observed patterns.

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