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Phys222 Electric Fields for Continuous Charge Distributions Electric Fields for Continuous Charge Distributions Names: Grade: ____________ This is a group write-up. Notes on your report: You have been given six problems. Follow the write-up criteria shown below. There is no need to word- process these problems but each problem should have separate pages, i.e., be sure to start a new page when you start a new problem. Work in groups. You should hand in one report but it is not appropriate to divide and conquer entirely. Each group member must participate in the solution of each problem. Each person will be responsible for the final write-up of at least one problem. You must understand and be able to reproduce the answers to all of your groups’s problems. Write-up Criteria (Be neat and professional.) 1. Restate the problem. 2. Include a drawing of the charge distribution and the point of interest. 3. Include a sample vector diagram, showing the electric field vectors for critical dq’s. 4. Use the diagram in point 2 above to help explain any cancellations due to symmetry. 5. Set up the integrals to determine the non-zero electric field components. Show your work/reasoning for building the integrals. 6. Solve the integrals. Box or underline your solutions. 7. Assess: is your result reasonable? Do the solutions have the correct behavior far away from the charge? Note: It is recommended that you use lots of scratch paper to work through the problems. However, when you turn in a draft next week or the final version in two weeks, your work must be very clear and neat, else I won’t be able to assist you (draft) or there will be an automatic deduction (final version). Set extra time aside for a clean write-up, it does take some time. Think first, then commit to paper. Don’t skip steps in your algebra: if I don’t see your work there will be a deduction. Phys222 Electric Fields for Continuous Charge Distributions Problem 1: As shown in the figure, a non-conducting rod of length L has charge q uniformly distributed along its length. a. What is the linear charge density of the rod? b. What is the electric field at point P, a distance a from the end of the rod? c. If P were very far from the rod compared to L, the rod would look like a point charge. Show that your answer to part b reduces to the electric field of a point charge for a ? L . Phys222 Electric Fields for Continuous Charge Distributions Problem 2: A “semi-infinite” non-conducting rod (that is, infinite in one direction only) has uniform linear charge density . Show that the electric field at point P makes an angle of 45⁰ with the rod and that this result is independent of the distance R. (Hint: Separately find the parallel and perpendicular (to the rod) components of the electric field at P, and then compare those components.) Phys222 Lab Activity 3: Electric Fields for Continuous Charge Distributions Problem 3: A thin glass rod is bent into a semicircle of radius r. A charge q is uniformly distributed along the upper half, and a charge q is uniformly distributed along the lower half, as shown. Find the magnitude and direction of the electric field E at P, the center of the semicircle. Phys222 Lab Activity 3: Electric Fields for Continuous Charge Distributions Problem 4: A non conducting hemisphere of radius R centered at the origin has a total charge Q spread uniformly over its surface. The hemisphere is oriented such that its base is in the (y,z) plane. Find the electric field anywhere along the x axis for x > 0. Give explicitly the value of the electric field at x = 0. Hint: consider the hemisphere as a stack of rings. y x z Phys222 Lab Activity 3: Electric Fields for Continuous Charge Distributions Phys222 Lab Activity 3: Electric Fields for Continuous Charge Distributions Problem 5: A thin, non-conducting rod is in the shape of a semicircle of radius R. It has a varying charge per unit length described by 0 sin 2 , where is defined in the figure. a. Sketch the charge distribution along the semicircle. b. What is the direction of the electric field E at point ), the center of the semicircle? c. Find the magnitude of the electric field at point O. Phys222 Lab Activity 3: Electric Fields for Continuous Charge Distributions Problem 6: a. Consider a uniformly charged thin-walled right circular cylindrical shell having a total charge Q, radius R, and height h. Determine the electric field at a point a distance d from the right side of the cylinder as shown in the figure. (Suggestion: treat the cylinder as a collection of ring charges.) b. Consider now a solid cylinder with the same dimensions and carrying the same charge, uniformly distributed through its volume. Find the filed it creates at the same point.