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Electric Field Due to a Uniformly Charged Ring
Developed by A. Titus
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.
Exercises
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.
ring-side
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.
2.
3.
4.
5.
Sketch a picture showing the four point particles (representing the pieces), the point P
where the electric field is being computed, and the ring.
What is the charge of each piece?
What is the electric field due to each piece at the given point?
What is the net electric field at the given point?
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.
2.
3.
4.
𝑁=4
𝑁 = 10
𝑁 = 100
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.
ring-side-2
Try the following values of 𝑁.
Compute the net electric field for the following values of 𝑁.
1.
2.
3.
4.
𝑁=4
𝑁 = 10
𝑁 = 100
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.