Electric Field Due to a Line of Charge
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LivePhoto Physics Activity 27 Name: ______Page 1 of 4 Electric Field Due to a Line of Charge
Consider a thin insulated rod that carries a known negative Qrod y charge Qrod that is uniformly distributed. It is possible to determine the electric field along a line perpendicular to the rod that passes through its center using the following equation – x often derived in introductory texts: r kQ L theory rod ˆ Qball Erod i Test 2 2 [Eq. 1] r r (L / 2) Charge where L is the length of the charged part of the rod, r is the distance from the test charge to the center of the charged part of the rod, and Qrod is its total charge. The constant k is the well- known Coulomb constant. Figure 1: A uniformly charged rod exerts a force on a test charge In this activity you will examine a digital movie of a charged rod exerting a force on a hanging “test charge” along with a Logger Pro analysis to determine if the theoretical equation [Eq. 1] describes the relationship between r, L and meas the measured electric field, Erod at the location of the test charge.
How a Horizontal force on a Hanging Ball Displaces It: Before doing this activity you should have completed Activity 25 entitled Coulomb’s Law for Two Charged R Spheres. There you found that if the center of a small ball of mass m is hanging vertically a distance R from a point of
support and then becomes displaced by a distance x < 1. Preliminary Questions Note: You will receive full credit for each prediction made in this preliminary section whether or not it matches conclusions you reach in the next section. As part of the learning process it is important to compare your predictions with your results. Do not change your predictions! As you proceed with this assignment, you’ll be working with a short video clip entitled Physics with Video Analysis 27 - 1 Name: ______Page 2 of 4 (b) Sketch electric field vectors at the location of a positive test charge for the three small segments (labeled 1, 2, and 3) based on its interaction with the negatively charged rod shown in the Figure 3 below. Please indicate both the direction and relative magnitude of each of your vectors. Base your scale on the vector associated with segment 2 and assign it a length of 2 cm. Hint: Assume each segment behaves like a point charge so you can use Coulomb’s Law to determine the relative magnitudes of the vectors. Figure 3: A uniformly charged rod exerts a force on a test charge (c) Find the direction of the E-field at the location of the test charge. Turn your thoughts back to the movie. Assume that the center of the charged part of the rod shown in the movie (i.e., between its bottom end and the bottom of the hand) lies on the x-axis. What will the direction of the net electric field be at the negatively charged hanging ball’s location if the charged rod, like that shown in Figure 3, acts on it? Use the x-y coordinate system shown in Figures 1 and 3 to describe the direction of this E-field. Note: For now, ignore the fact that center of the rod in the movie is not quite on the x-axis. (d) Equation 1 predicts that the E-field lies along the x-axis only. Does this agree with the conclusion you reached? Why or why not? (e) Assume the hanging ball acts as a “test charge” by carrying an excess charge, Qball. If its center is displaced from its original vertical axis by a distance, x, write down the equation need to calculate the electric field at the ball’s center based on measurements of the ball’s horizontal displacement, x, its mass, m, charge, Qball, the gravitational constant, g, and the distance, R, from the ceiling to the center of the ball (with x << R). meas Erod,x [Eq. 3] 27 - 2 Physics with Video Analysis 2. Activity-Based Questions In this section, you will be working with a Logger Pro file entitled (a) Determine the length of the charged part of the rod: Use the Photo Distance tool ( ) to find the length L of the rod (between the rod’s bottom and the bottom edge of the hand). Enter your measured value in the space below. L = m (b) Determine whether the calculated values of the electric field based on measurements are consistent with meas theory: The key to this determination is to see if the Logger Pro graph of Erod,x vs. r (based on Equation 3) can be fit by the function described in Equation 1. To do this fit: (1) Choose Curve Fit in the Analyze menu. (2) Scroll down through the equations and select the one that looks like Equation 1 (V-rod_Pt Charge Theory). (3) Do the fit and draw the fit line in the graph. Enter the values of A (to four significant figures) and L to two significant figures) on the graph to the right. Best Fit: A = ______N•m2/C L = ______m (c) Comment on whether you think the fit is good enough to verify that the theoretical equation adequately describes the electric field due at various distances, r, along a perpendicular axis passing through the center of the rod. Physics with Video Analysis 27 - 3 Name: ______Page 4 of 4 (d) Note that, to within a sign, the A in your fit equation corresponds to the product of the Coulomb constant ( 9 2 2 k 8.9910 N m /C ) and the charge on the rod. Use these facts to calculate the charge on the rod, Qrod in nC (nanocoulombs). Show your calculations and express your answer using three significant figures. 3. Reflections on Your Findings (a) Given the methods used to charge the rod and the hanging ball, do you expect the charge on the rod to be greater than the charge on the ball or less than the charge on the ball? Why? (The 35 nC listed for the ball is based on measurements made with the charge sensor distributed by Vernier Software & Technology.) (b) In light of your answer in part (a), comment on how charge you calculated for the rod compares with the 35 nC charge measured for the hanging ball. (a) You should have noticed that the “fit value” of the rod length L is almost 10% lower than the directly measured value. Hint: What impact might the rod being a bit off center have of the experimentally determined electric field value? 27 - 4 Physics with Video Analysis