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Electric Field Due to a Uniformly Charged Ring
Developed by Aaron Titus  Published July 17, 2016
This set of exercises guides students in calculating the electric field at locations around a charged ring. Students will compare numerical results to an analytic solution for points along the axis of the ring, but they will also compute electric field at any point around the ring.
Subject Area  Electricity & Magnetism 

Level  First Year 
Available Implementation  IPython/Jupyter Notebook 
Learning Objectives 
1. analytically calculate the electric field along the axis of a charged ring. (Exercise 1)
2. numerically compute the electric field due to a charged ring at points on the axis of the ring. (Exercises 23)
3. numerically compute the electric field due to a charged ring at any point in space. (Exercises 4)
4. explain why the computation of electric field at points along the axis of a charged ring does not depend on the number of point particles used to model the ring. (Exercises 3)
5. explain why the accuracy of the computation of electric field at points not on the axis of the ring depends on the number of point particles used to model the ring. (Exercise 4)

Time to Complete  120 min 
These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).
### 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.
![ringside](./images/efieldring/ringside.png)
### 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 rerun 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.
![ringside2](./images/efieldring/ringside2.png)
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.
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Credits and Licensing
Aaron Titus, "Electric Field Due to a Uniformly Charged Ring," Published in the PICUP Collection, July 2016.
The instructor materials are ©2016 Aaron Titus.
The exercises are released under a Creative Commons AttributionNonCommercialShareAlike 4.0 license