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## Visualizing Curl: Instructor's Guide

### Main Ideas

• curl as the circulation per unit area around an infinitesimal loop
• sign and relative magnitude of curl at various points in space
• checking predictions with Maple/Mathematica

Estimated Time: 20 min

Students view several vector fields and calculate their curl in order to get a sense of what a field with a non-zero curl looks like.

This worksheet is designed to be an instructor-led activity. You would need to add appropriate instructions and questions to use this as an independent student activity. The activity can be used quite effectively with the instructor projecting the worksheet at the front of the room if students do not have access to a computer for each small group.

### Prerequisite Knowledge

• Familiarity with Maple
• Vectors
• Algebraic understanding of curl

### Activity: Introduction

We precede this activity with a derivation of the rectangular expression for curl from the definition that (the magnitude of a particular component of) curl is the circulation per unit area around an appropriately chosen planar loop. Our derivation follows the one in “Div, grad, curl and all that”, Schey, 2nd edition, Norton, 1973, p. 74.

This worksheet shows a number of different vector fields. Most vector fields are shown as a cross-section of the field and it is assumed the the vector field is independent of the third (unshown) dimension. Students are asked to use the definition of curl as the circulation per unit area around an infinitesimal loop to predict the direction and relative magnitude of the curl at various points in the vector field. The worksheet then calculates the curl, so students can check their predictions.

### Activity: Student Conversations

• Symmetry: Students should be encouraged to see that it is easier to choose a loop that respects the symmetries of the vector field, i.e. pineapple chunks for cylindrical fields, etc.
• Various Points: Make sure to look at several different points in space for each vector field, not just the origin. Use this to emphasize that curl is itself a field which, when combined with Ampere's Law, tells you what the current density is at each point in space.
• Ways to get zero curl: It is a good idea to have groups do an example of zero curl where all the legs in the circulation around the axis are zero and one where it is zero because of cancellation of 2 legs.
• Positive or negative curl: Students should should see that, for the vector field that circles the origin, different length scalings lead to different signs for the curl, depending on whether they are adding larger vectors along the longer length arc or smaller vectors along the longer length arc. The last example $\left(\frac{1}{r} \hat{\phi}\right)$ can be framed as a “Jeopardy question” where they are asked to discover which scaling leads to zero curl everywhere except at the origin.
• The ANSWER is: a nontrivial field that looks like the one on the screen which has zero curl everywhere but the origin.
• What is the QUESTION? What is the magnetic field around a current carrying wire? Nature picks out this special case.
• It is subtle (with the Delta function curl!) and surprising for students, so it is often worth talking/working through the origin and non-origin cases separately for the vanishing of a curl that “looks curly” nearly everywhere .

### Activity: Wrap-up

No particular wrap-up is needed.

### Extensions

This activity pairs nicely with the Visualizing Divergence activity.

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.
• 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.

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