Voltage Plotting Lab Report

Torin Kilpatrick, John Hoffman Potential and Electric Fields Lab Report Introduction: The purpose of this lab was to gather enough data to create an accurate representation of the electric field in three separate systems. In one of the systems involving concentric circles, it was also necessary through data analysis and critical thinking to decide which theory applied to the system and why. Lab Set-up: The experiment was set up by connecting a power supply to a conducting sheet, creating an electric field. A voltmeter was used to determine the electric potential at each point marked on the conducting sheet. This was done by holding the high potential cord of the voltmeter (indicated by the color red) on the thumb tack used to secure the high potential wire for the power source, and moving the black wire of the voltmeter to various points on the conducting paper which caused the electric potential at the black wire to be displayed. A layout of the experiment is shown below.

Three trials were conducted using different silver paint designs on each one. Each design, shown below, created a different electric field.

Each charged sheet was marked with a grid of points, and we measured the electric potential at each of these grid points. The first sheet (sheet A) simply had two dots with a wire of high potential attached to one and a ground wire attached to another. The second sheet (sheet B) had two parallel lines marked on it that had points between them and outside of them. These lines would be connected separately to the power supply with the grounded wire on the bottom line. The final sheet (sheet C) contained a dot surrounded by a circle with the outside circle connected to the ground wire and the high-potential wire connected to the dot in the center. Theory: After data points for each sheet were collected, an image of the electric fields could be crafted. Equipotential lines indicate places which have the same electric potential. For this reason they are perpendicular to electric field lines, which indicate changes in electric potential. For the system in sheet A, there are two point-like locations—one with high potential, and one with low potential. This potential difference creates a field with electric field lines running from the point of high potential to the point of low potential (the upper point to the lower point). For the system in sheet B, the two lines of different potential (the upper being the higher potential and the bottom being the ground) create an electric field with field lines running primarily from the upper line to the lower line in straight lines, slightly curved at the ends of the lines. For the system in sheet C, the theory differs depending on whether or not the circle and dot represent a cross section of a cylinder or a cross section of a sphere. For

both systems, since the ground is placed on the outer circle, there will be no electric field outside of the circle, and the location of greatest potential will be at the center dot inside the outer ring. If the system is a cylindrical cross section, the electric field would be determined by the function (λ/(2πεor)), but if the system is a spherical cross-section, the electric field would be determined by (kQ)/r. This means, if the system is a cylindrical cross-section, the potential will weaken logarithmically (as a function of (highest potential) – (λ/(2πεo))ln(r))) but if it is a spherical cross section, the potential will weaken as a function of (highest potential) – kQ / r. The electric field was dependent upon the potential difference between the electrodes since a stronger potential difference creates a stronger electric field. Lab Results: 3D graphical representations of the electric potential as a function of distance were created using Excel’s 3D graphing capabilities and are shown below. 3D Potential Graph of Electric Potential as a Function of Radius on Sheet A

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24 23 22 21 20 19 18 17 16 15 Electric 14 13 Potential (in 12 11 10 Volts) 9 87 6 5 43 2 1 0

3D Graph of Electric Potential as a Function of Radius for Sheet B

24 23 22 21 20 19 18 17 16 15 Electric 14 Potential (in 13 12 11 10 Volts) 9 8 7 6 5 4 3 2 1 0

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3D Graph of Electric Potential as a Function of Radius for Sheet C

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24 23 22 21 20 19 18 17 16 15 Electric 14 13 Potential (in 12 11 10 Volts) 9 87 6 5 43 2 1 0

Equipotential lines can easily be created using Excel’s 3D graphing capabilities and changing the 3D perspective and rotation settings until the view is topographic. The areas indicated by different colors depict areas which have approximately the same potential.

Topographic View of Electric Potential in Sheet A 1 4 7 10 13 16 19 22 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

Topographic View of Electric Potential in Sheet B 1 4 7 10 13 16 19 22 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

Topographic View of Electirc Potential in Sheet C 1 4 7 10 13 16 19 22 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

Lab Analysis: The electric field lines match almost perfectly with the predictions in the theory section. The limits of the theory lie in the fact that there are an infinite amount of electric field lines but a limited amount of space to draw them on paper, so the accuracy of the electric field drawing is limited. Also, small inconsistencies, especially evident in the circular sheet (C), illustrate that there might be imperfections in the conducting sheet (slight nonuniform resistance) or slight inaccuracies in the voltmeter. The theory predicts the behavior of electromagnetism in an ideal setting, but the real-world is not an ideal place. The concentric circles sheet best represents the cross section of a cylinder. This makes logical sense, because the conducting silver paint has a slight amount of thickness making the painted system seem very slightly cylindrical. This theory is also backed up by the data. If our predictions are correct, then the potential difference should change as a function of 23.4 – (constant) ln(r). We find that this is true. Taking the center row of data and plotting that as a 2-dimensional function yields a graph that is most consistent when the constant for the function is approximately 12 (which would mean that the linear charge density is ~6.7(10)-10 C/m). Conclusion: This lab provided an extremely beneficial visual representation of the concepts learned in the past few weeks. It was also cathartic, but besides that it caused us to understand, point by point throughout the conducting paper, how the potential difference changed, and later, how the whole map of different levels of potential difference looked in 3D and in 2D. Being required to describe and draw the relation of the electric field lines to the equipotential lines reinforced the understanding of the connections and the differences between electric field and electric fields.