Partnership for Integration of Computation into Undergraduate Physics

### Exercise 1: Computing Electric Field Along the Axis of a Charged Ring Analytically A uniformly charged ring of radius $R=0.02$ m has a charge of $Q=1$ nC. What is the electric field at a distance $R$ from the center of the ring, along the axis of the ring? A side view of the ring is shown below. The red sphere represents the point $P$ in space where you are calculating the electric field.  ### Exercise 2: Computing Electric Field Along the Axis of a Charged Ring Numerically By Hand To numerically compute the electric field, break the ring into small equally spaced pieces and treat each piece as a "point particle." Compute the electric field due to each piece and use superposition to find the net electric field. Suppose you break the ring into only four equally spaced pieces (each piece being a quarter of the ring). 1. Sketch a picture showing the four point particles (representing the pieces), the point P where the electric field is being computed, and the ring. 1. What is the charge of each piece? 2. What is the electric field due to each piece at the given point? 3. What is the net electric field at the given point? 4. How does your answer compare to your answer in Exercise 1? How can we improve the accuracy of the numerical solution? ### Exercise 3: Computing Electric Field Along the Axis of a Charged Ring With a Computer Write a computer program to compute the net electric field at the given point in space. Follow the same technique you used in Exercise 2. However, your code should compute the position of each piece based on the total number of pieces $N$ so that you can easily change $N$ and re-run your program. Compute the net electric field for the following values of $N$. 1. $N=4$ 2. $N=10$ 3. $N=100$ 4. $N=1000$ Compare your results to the analytic result in Exercise 1. Comment on any interesting pattern you might notice. Explain the reason for the pattern you observe. ### Exercise 4: Computing Electric Field Due to a Charged Ring at Any Point in Space Now, compute the electric field at a point $2R$ from the center of the charged loop, in the plane of the loop, as shown below.  Try the following values of $N$. Compute the net electric field for the following values of $N$. 1. $N=4$ 2. $N=10$ 3. $N=100$ 4. $N=1000$ Does the result depend on $N$? Compare to what you found for Exercise 3. Explain why the variation with $N$ is different for Exercises 3 and 4.