[BUG] curl question

[Corinne A. Manogue]

Let me add to the discussion.  If you put the paddle wheel in somewhere,
not at the origin, then there are long arrows pushing on the paddle
closest to the origin and short arrows pushing on the paddle furthest from
the origin, so at a first guess, you would expect the paddle wheel to
spin.  But there is a sense in which there are "more" arrows further from
the origin.  So more short arrows cancel fewer long arrows.  It is hard
with the paddle wheel picture, to see how this goes quantitatively, so I
also like to show the students the integral of F.dr around a closed loop.
(We actually use this integral to define the curl, teaching the
combination of paddle wheels and line integrals is quite effective if you
have the class time to do both).  So, the integral of F.dr around an
"infinitesimal"  closed loop is proportional to the curl at that point.
You can get a reasonable idea of what is happening by taking a finite
closed loop.  In principle, any closed loop will work, but if you pick a
nice shape, you can often "see" what is happening best.  For this problem,
pick a pie shaped piece with the pointy end cut off in a circular arc
(i.e. a big circular arc, a smaller circular arc closer to the origin, and
two radial line segments that join the arcs).  Now, F is perpendicular to
the radial line segments, so integral F.dr for those portions of the curve
is zero.  On each of the circular arcs, F has constant magnitude and is
parallel to the curve, so the line integral for these parts is the
magnitude of F times the length of the arc.  If the vector field falls off
fast enough (magnitude gets smaller further from the origin), then the
contribution from the big arc (smaller F, but larger arclength) is smaller
than the contribution from the small arc (larger F, but smaller arclength)
and the curl will have one sign.  This is what I meant by "more" arrows
farther from the origin.  If the fall off is too slow, then the
contribution from the larger arc will dominate and the curl will have the
opposite sign.  For the particular fall off that you chose (~1/r), it
turns out the contributions from the two arcs exactly cancel and the curl
is precisely zero everywhere except at the origin.  What fascinates me is
that the universe tends to pick out those cases which are balanced in this
way, in this case for the magnetic field outside a current carrying wire.
You get a similar thing happening for the divergence outside of a point
charge.

Corinne