## Student handout: Equipotential Surfaces

Students are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials.
• group Small Group Activity schedule 120 min. build Quadrupole plastic surface set, small group whiteboard description Student handout (PDF)
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What students learn
• Add the potential due to each charge to calculate the potential due to a collection of charges.
• Equipotential surfaces are 3D surfaces where the potential is a constant value.
• The spacing between equipotential surfaces, by convention, is such that the change in potential is the same for adjacent equipotential surfaces.
• Therefore, close spacing means the potential is changing quickly with distance; wide spacing means the potential is changing slowly.
• Considering equipotential surfaces is only one of many ways to visualize the electric potential in space.
• Inverse square force law means that the potential changes faster closer to the source---far away, the potential changes slowly.

Start with a Simpler Case: The electrostatic potential due to a particle with charge $q$ is: $V(r)=\frac{kq}{r}$

where $k$ is the electrostatic constant and $r$ is the distance from the particle.

On your whiteboard, identify all the points with the same value of potential around a single point charge. Repeat for several different values of potential.

• What shapes have you drawn?
• If you wanted the difference in potential represented by the shapes to be equal, how are they spaced?

Add Complexity: Draw equipotential surfaces for the potential due to 4 particles with equal, positive charge arranged in a square.

Examine a New Case: Repeat for a quadrupole: 2 positively charged particles and 2 negatively charged particles arranged in a square, with “like” charged particles on opposite corners.

Extend to New Surfaces: The red surface represents the potential of a quadrupole in the plane of the charges (at $z=0$ cm). What would the potential look like in the $z=1$ cm plane? What would be different? What about the $z=-1$ cm?

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