• Conceptual understanding of flux
• Finding the component of a field perpendicular to a surface
• Finding the differential area element of a surface

Estimated Time: 30 minutes

Prompt: Find the flux through a right cone of height $H$ from the vector field $\Vec{F} = C\,z\,\hat{k}$ (see reflections for a discussion of this vector field choice).

This prompt is open-ended in that it doesn't specify either the location of the cone or whether or not the circular top of the cone is to be considered part of the surface. We like to leave it open-ended, see what students do, and when students question the open-endedness, give a mini-“sermon” on the ill-posedness of most real world problems. If you are short of time, or otherwise want to avoid these questions, you should use a more explicit prompt.

If you choose the point of the cone at the origin (and allow it to open upward, like an icecream cone), then the problem can be solved in spherical coordinates as well as the obvious cylindrical coordinates. This is an interesting example of the fact that it is often easier to do surface integrals with cylindrical symmetry in spherical coordinates.

Students should be familiar

• with integrating over a surface
• 3-D vector fields
• finding the differential area element of a surface

• Tabletop Whiteboard with markers

Short lecture introducing the concept of flux (as the amount of a vector field perpendicular to a surface) and how to calculate it: $$ \Phi = \int_S\, \Vec{F}\, \cdot \,d\Vec{A}$$

• Choice of coordinates - some groups will choose cylindrical and others will choose spherical coordinates and it can be done with either. Occasionally, a group will attempt to use Cartesian coordinates, but they usually realize quickly that this is not a good choice.
• Into or Out of the Cone? - the sign of the flux depends on whether we want to calculate the flux up through the cone or down through the cone. Students need to be aware of this choice.
• Evaluating the field on the surface - in this case, the vector field varies along the surface of the cone. If the vector field were constant, student could find the projection of the surface on the x-y plane. In this case, however, student must integrate over the surface. A nice extension might be to consider a closed cone and ask them to find the total flux (in this case, the function will be constant over the base).
• Finding the differential surface element - students can find the differential surface element by taking the cross product of two $d\Vec{r}$ vectors lying in the plane (normalized appropriately). Several issues for the students arise, such as
  • writing down the $d\Vec{r}$'s, and
  • choosing the direction of the area element (i.e. which order of the vectors in taking the cross product).
• Limits of integration - many students have difficulty determining the limits of integration.

We do a brief summary of the main points to wrap up the activity.

• This is also good place to talk about the affordances of different choices for coordinates (e.g. ask a group that solved it in cylindrical and one that solved it in spherical to compare).
• It is important to reinforce the method of constructing the $d\vec{a}$ vector by crossing $d\vec{r_1} \times d\vec{r_2}$, especially if some groups instead tried to look at the areas geometrically and craft an expression.

This activity is part of a sequence of activities on the Geometry of Flux.

• Preceding activities:
  • Concept of Flux: A kinesthetic activity in which students use rulers to represent a vector field while the instructor uses a hula hoop to represent a surface which is followed by a class discussion on what contributes to the flux through a surface. There is also a narrative of this activity that describes it in detail for a specific class.

• Follow-up activities:
  • Visualizing Flux: A Mathematica (or Maple) activity which allows students to work in small groups to explore the flux of the electric field due to a point charge through the surface of a cube, a Gaussian surface.
  • Gauss's Law: A compare and contrast activity in which students are asked to work in groups to find the electric field using Gauss's law, in integral form, for either a spherically or cylindrically symmetric charge density.