# Potential Due to Point Charges

## Prerequisites

Students will need to know some VERY basic Maple. The following Maple worksheet may be helpful. - links

## In-class Content

• Refresher 2 (15 min)
• Equipotentials Activity (85 min)
• Lecture before the activity:
1. SWBQ: Write down something you remember about superposition.
2. How would you use superposition to find the gravitational potential halfway between the Earth and the Moon?
3. Both electrostatic potentials and gravitational potentials satisfy the superposition principle. It would have been very difficult for us ever to have developed the field of physics if this were not the case.
• Draw equipotential lines for a configuration of four positive point charges arranged in a square.
• Whiteboards should be prepared with four points with the correct spacing.
• Students may not know what an “equipotential” is or what it means.
• Let the instructors do as much direct talking with the groups as possible before whole-class discussion.
• Whole-class discussion (middle of activity):
1. Make sure you pay attention to how far apart your equipotential lines are.
2. What do the equipotential surfaces look like for a point charge?
• Are the proper equipotential surfaces two-dimensional or three-dimensional?
3. How do you represent equipotential lines with different values?
• Let students go again until most have good answers, then bring back together for wrap-up of the correct solution plus introduction to Mathematica.
• Draw equipotential lines for a quadrupole.
• Warn students not to erase the dots.
• Hand out the surface at the start of the activity. (Students won't know that they are dry-erasable!)
• Tell students they can modify the Mathematica worksheet to show this charge configuration.
• Whole-class wrap-up (end of activity):
2. What is the meaning of the spacing between equipotential lines?
3. What was different about the two charge configurations we did?
6. Submit a reflection on Canvas!

## Homework for Static Fields

1. (V4ChargesSquare)

Four point charges sit at the corners of a square in the $xy$-plane. A positive point charge is located at $(a,a,0)$ and another is located at $(-a,a,0)$. A negative charge is located at $(-a,-a,0)$ and another is located at $(a,-a,0)$.

1. Find the electric potential at any point $(x,y,z)$.

2. Is the $yz$-plane an equipotential surface? Explain. If so, what is the value of the potential?

3. Is the $xz$-plane an equipotential surface? Explain. If so, what is the value of the potential?

4. Do you expect any other equipotential surfaces to exist? Explain. If you do expect one or more equipotential surfaces, use the Mathematica function ContourPlot3D to plot one. All plots should include a title, axis labels and a legend if appropriate. Note: providing a plot does not count as an explanation for why you would expect an equipotential surface to exist.

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