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## Visualizing Electric Flux: Instructor's Guide

### Main Ideas

- use of Maple/Mathematica to calculate electrostatic flux
- verifying the integral form of Gauss's Law.

### Students' Task

*Estimated Time: 20 min*

To explore the concept of flux by seeing how it is calculated using Maple/Mathematica.

Students can explore the effects of putting a point charge at various places inside, outside, and on the surface of a cubical Gaussian surface. First the worksheet shows the electric field for the charge, then calculates the value of the flux integrand on the top surface of the box, the value of the integral on the top of the box, and the value of the integral over the whole cube.

### Prerequisite Knowledge

- An introduction to flux integrals. See, for example:
- GVC § Flux

- Many students have seen the integral form of Gauss's Law in their introductory physics sequence, so the overall result is not a surprise. You may choose to remind students about Gauss's law before the activity, or leave students to discover Gauss's law by the end of the activity.

### Props/Equipment

### Activity: Introduction

We usually talk the students through the worksheet with the charge at the center of the box and then encourage small groups to try putting the charge in other places.

### Activity: Student Conversations

**Examining the integrand:**With the charge at the center of the box, ask students why the value of the integrand is largest is the center of the top surface of the box. Bring out the fact that the charge is closer to that point of the surface**and**that the entire electric field vector is perpendicular to the surface there. Draw a picture showing these two aspects of the geometry.

**Understanding more about how Maple/Mathematica works:**When students are exploring putting the point charge at other places, point out that sometimes Maple/Mathematica can do the flux integrals exactly (often in terms of complicated expressions involving arctangents). (The “evalf” command in Maple is useful in these cases.) At other times, Maple/Mathematica will do the integral numerically. Point out where the worksheet sets various constants to one, so that the integral can be done numerically. Also point out the round-off errors that occur.

**When the charge is not in the center:**When encouraged to explore the consequences of putting the point charge at a variety of positions, many groups will choose points on a face, edge, or vertex of the cube. The Maple/Mathematica code is robust enough to handle these situations, yielding $q\over 2\epsilon_0$, $q\over 4\epsilon_0$, and $q\over 8\epsilon_0$, respectively. A few students can be bothered by the idea that an infinitesimal point charge can be partially inside the box and partially outside the box. For these students, returning to the idea of flux and drawing pictures of how much of the electric field points through a side of the box (or is parallel to a side of the box) can be helpful. Electric field lines can also be a helpful representation.

**Choosing a point:**When students are choosing a particular point, a few may choose to insert equations into the coordinate for the point. Depending on their choice, this may not be a single point if their variables are not clearly defined. For these students, it may be helpful to ask them the exact location of their point with respect to the cube.

### Activity: Wrap-up

If not already addressed, students should be alerted to the relationship between electric flux and the charge enclosed by the surface (namely, Gauss's Law).

### Extensions

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.
- Drawing Electric Field Vectors: In direct analogy to Drawing Equipotential Surfaces, this small group activity has students sketch the electric field due to a quadrupole in the plane of the charges.

- Follow-up activities:
- 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.

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.
- Calculating Flux: A small group activity in which students calculate the flux of a simple but non-constant vector field through a cone.

- Follow-up activities:
- 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.