Graphing Voltage vs. Position, Voltage vs. Radius, and Electric Field vs.

Graphing Voltage vs. Position, Voltage vs. Radius, and Electric Field vs.[Type the document subtitle]

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Results and AnalysisWe began the lab by drawing two circles with the conducting paint on conducting paper, one with radius a and the other with radius b, with b greater than a. We connected the outer circle with radius b to the ground voltage terminal of the battery eliminator (so Vb is zero). Our setup is shown below. (insert diagram)We then Gausss Law to derive an expression for the electric field in each section of the paper: outside of the larger circle, between the two circles, and inside the smaller circle. We represented each circle on the paper as a very small cylinder, and chose our Gaussian surface in each of the three cases to be a cylinder of radius. Our derivations are shown below, along with the Gaussian surface for each case.Case 1: r a(insert diagram)(insert derivations, use old computer, split in two columns)Case 2: a r b(insert diagram)(insert derivations, use old computer, split in two columns)Case 3: r b(insert diagram)(insert derivations, use old computer, split in two columns)We also know that the relationship between voltage and electric field is given by:(insert equation)We can then use this relationship to find the linear charge density .(insert derivation)We then substitute this expression for into each of our equations for the electric field so that our equations depend only on the radius of each circle, the radius of the Gaussian surface (in other words, the distance to the point you are measuring from the center of the circle), and the emf of the battery.We then once again used the equation (insert equation)and our electric field equations to find the voltage V at any point in any of the three areas on the paper (when r a, when a r b, and when r b).For the first case, when r a, we chose the initial point 1 to be at r, and the second point to be along the inner circle.(insert derivation)For the second case, when a r b, we had the first point be located at r and the second point on the edge of the outer circle.(insert derivation)For the last case, when r b, the initial position is outside the circle and second position is on the outer edge of the larger circle.(insert derivation)Once we had found expressions for the voltages in the three areas, we proceeded to test these values experimentally by using the voltmeter to test the voltage at each point on the grid on the conductive paper.(insert table)We then used the data we had found to construct several different graphs in Excel. For these graphs, we had to use the values of the radii a and b and the emf of the battery . We measured a, b, and to be the values shown below.(insert values)The first one we made was an electric potential map. We did this by having the xy-plane be the position of the point on the paper and the z axis represent the values of the voltage at each point. Our voltage values were the experimental values we took previously, and we know that the distance between each point on the grid is 1 cm, or .01 m.(insert graph)As you can see, this graph corresponds to our theoretical expressions for the voltage. In the xy-portion of the graph where the point measured is outside the radius of the larger circle, the voltage is zero. In the portion of the xy-plane where the point measured is between the radii of the two circles, the voltage drops down exponentially as the radius increases. When the point measured is inside the inner circle, the voltage stays constant at about 4.5, which is the emf of the battery eliminator.The next type we created was a graph of the potential vs. the radius. We made two different graphs for this type; one using the experimental values of the potential and one using the theoretical values of the potential. We took the values from the point where r = 0 to the last point on the graph, along the y-axis so that it would be easy to find values for r, since each grid point was spaced .01 m apart.(insert data)(insert graph)As you can see, our experimental graph matches up fairly closely with our theoretical graph. When the radius is less than or equal to the radius of the inner circle, the potential is constant at around 4.5 volts, the emf of the battery. When the value of the radius is between the values of the two circles radii, the voltage drops off like an exponential graph. When the value of the radius is greater than value of the radius of the outer circle, b, the voltage is zero.Our final graph was a graph of electric field vs. radius. We found the values of the electric field using the formula and approximation(insert equation)Each r would be .01. However, because this approximation is over the entire interval, the halfway point between each r interval is chosen as an average. We will once again use the voltage values that we obtained experimentally and the voltage values we obtained theoretically to get two different graphs. The experimental voltages and radii are once again measured along the y-axis, and the theoretical values are obtained using the equations we derived for voltage.