Students are tasked to find what electric field looks like from a quadrupole.

Estimated Time: 20 - 30 minutes

• How the gradient acts a potential field

• What the potential field looks like for a quadrupole

• $\Vec E = -\Vec \nabla V$

• Tabletop Whiteboard with markers

Ask students to sketch the electric field vectors of a quadrupole. Students should familiar with the relationship between the electric field and potential, $\Vec E (\Vec r) = -\Vec \nabla V (\Vec r)$.

Students tend to use two main approaches to this problem which are as follow:

• Vector addition of the electric field from each charge

• Using the gradient on the known potential field of a quadrupole

Students may try inadvertently draw electric field lines instead of electric field vectors. Drawing electric field lines in two dimensions will not show the appropriate electric field drop off. Electric field lines in three dimensions are required to see the correct drop off. Some students will draw electric field vectors with the correct direction, however, will not consider the importance of the magnitude of the vectors.

Some students struggle with the electric field vectors at points between the charges. Discussing the potential field of the quadrupole is helpful in guiding students to sketching the electric field vectors in the region between the charges.

This activity is part of a sequence of activities which address the Geometry of Vector Fields. The following activities are included in this sequence.

• Preceding activities:
  • Curvilinear Basis Vectors: This kinesthetic activity students are asked to point in $\hat{r}$, $\hat{\theta}$, $\hat{\phi}$, and $\hat{z}$ directions in reference to an origin within the classroom which begins class discussion regarding about the directions of curvilinear basis vectors at various points in space.
  • Visualizing Gradient: This activity uses Mathematica to show the gradient of several scalar fields.

• Follow-up activities:
  • Visualizing Electric Flux: This computer visualization activity uses Mathematica to explore the effects of placing a point charge inside, outside, and on a cubical Gaussian surface which allows students to visualize the electric flux of a point charge through a Gaussian surface in different locations with respect to the point charge.
  • Visualizing Divergence: This computer visualization activity has students predict the sign of divergence at various points in many vector fields generated by a Mathematica notebook.
  • Visualizing Curl: Similar to Visualizing Divergence, this activity uses a Mathematica notebook of various vector fields to assist students in the geometric interpretation of the curl of a vector field by predicting the sign of the curl at various points in vector fields.