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We proceed 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 length 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.

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.

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.

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. Near the end of this activity, they can be asked to discover which scaling leads to zero curl everywhere (except at the origin). This vector field represents the magnetic field around a current carrying wire. Nature picks out this special case.

**Rabindra Bajracharya

-Modified worksheet version of the curl activity

I started this activity by asking students how they would determine whether the curl of any vector field is zero or non zero. A student responded that he could find a curl by finding circulation per unit area. Then I wrote on board $$\text{Curl of}\, \vec{F} = \frac{\text{circulation}}{\text{tiny area}}$$ and asked if that was what he meant. Then I drew a constant vector field ($\vec{F} = \hat{y}$) on the board and showed how to find circulation (line integral) at a point in the field. I repeated the same illustration for a vector with non-zero curl ($\vec{F} = x\hat{y}$). Then I asked students to work in groups of 3-4 to determine whether the curl of the vector fields presented in the worksheet is zero or non-zero and ask to determine the signs of non-zero curls. The worksheet contained the graphical representations of the following four vector fields:

1. $\vec{F_1} = e^{-y^2}\hat{y}$,
2. $\vec{F_2} = e^{-x^2}\hat{y}$
3. $\vec{F_3} = -y\hat{x} + x \hat{y} = s \hat{\phi}$
4. $\vec{F_4} = - \frac{y}{(x^2+y^2)} \hat{x}+ \frac{x}{(x^2+y^2)} \hat{y} = \phi \hat{s}$.