• The conceptual and mathematical connections between electrostatic potential, electric field, energy, and force.

Estimated Time: 20 min

Students work in groups to create a concept maps connecting electrostatic potential, electric field, energy, and force.

• Concept maps
• Some familiarity with the concepts of electrostatic potential, electric field, energy, and force
• Gradients
• Vector line integrals

• Tabletop Whiteboard with markers

List the four quantities (electrostatic potential, electric field, energy, force) on the board and ask students to make a concept map with these central ideas. Students should be instructed that the arrows between these ideas should describe mathematically how to move between the quantities.

• Students will likely bring up the concept of work in their discussions, which will lead to a nice discussion of the connection between work and potential energy for a conservative field.
• Depending on how generally the prompt is pitched, students may bring up understandings of energy and force from their lower-division experiences (e.g. Newton's laws). We've found that this can lead to a very nice review of these topics and rich discussion of the coherence of physics.
• Students may forget about the minus sign between the gradient of the electrostatic potential and the electric field (and the gradient of the potential energy and the force). They will almost certainly forget about the minus sign going the other way: $\Delta V = - \int \Vec{E}\cdot d\Vec{r}$
• Students may continue to struggle with indicating vector and scalar quantities. This is an opportunity to emphasize clear and consistent notation for students who lack distinction between vectors and scalars.
• Some students will look up equations in their textbook or online without understanding the relationships between quantities.

Students should be directed to see the parallels between potential/field & energy/force, and field/force & potential/energy. A box illustrates these parallels nicely. Students should also be encouraged to see the connections between the mathematical formalism of going between these quantities and their conceptual understanding of the quantities (being able to “read” mathematics and translate into qualitative understanding).