b/The coordinate system
used in the following examples.
c/The fieldxˆ.
d/The fieldˆy.
e/The fieldxˆy.
Note that the curl, just like a derivative, has a differential divided
by another differential. In terms of this definition, we find Amp`ere’s
law in differential form:curlB=
4 πk
c^2jThe complete set of Maxwell’s equations in differential form is col-
lected on page 1026.11.4.2 Properties of the curl operator
The curl is a derivative.
As an example, let’s calculate the curl of the fieldˆxshown in
figure c. For our present purposes, it doesn’t really matter whether
this is an electric or a magnetic field; we’re just getting out feet wet
with the curl as a mathematical definition. Applying the definition
of the curl directly, we construct an Amp`erian surface in the shape
of an infinitesimally small square. Actually, since the field is uni-
form, it doesn’t even matter very much whether we make the square
finite or infinitesimal. The right and left edges don’t contribute to
the circulation, since the field is perpendicular to these edges. The
top and bottom do contribute, but the top’s contribution is clock-
wise, i.e., into the page according to the right-hand rule, while the
bottom contributes an equal amount in the counterclockwise direc-
tion, which corresponds to an out-of-the-page contribution to the
curl. They cancel, and the circulation is zero. We could also have
determined this by imagining a curl-meter inserted in this field: the
torques on it would have canceled out.
It makes sense that the curl of a constant field is zero, because
the curl is a kind of derivative. The derivative of a constant is zero.The curl is rotationally invariant.
Figure c looks just like figure c, but rotated by 90 degrees. Phys-
ically, we could be viewing the same field from a point of view that
was rotated. Since the laws of physics are the same regardless of
rotation, the curl must be zero here as well. In other words, the curl
is rotationally invariant. If a certain field has a certain curl vector,
then viewed from some other angle, we simply see the same field and
the same curl vector, viewed from a different angle. A zero vector
viewed from a different angle is still a zero vector.
As a less trivial example, let’s compute the curl of the fieldF=
xyˆshown in figure e, at the point (x= 0,y= 0). The circulation
around a square of sidescentered on the origin can be approximated706 Chapter 11 Electromagnetism