socurlE= cosˆzThis is visually reasonable: the curl-meter would spin if we put its
wheel in the plane of the page, with its axle poking out along the
zaxis. In some areas it would spin clockwise, in others counter-
clockwise, and this makes sense, because the cosine is positive
in some placed and negative in others.
This is a perfectly reasonable field pattern: it the electric field
pattern of a light wave! But Ampere’s law for electric fields says`
the curl ofEis supposed to be zero. What’s going on? What’s
wrong is that we can’t assume thestaticversion of Ampere’s law.`
All we’ve really proved is that this pattern is impossible as a static
field: we can’t have a light wave that stands still.
Figure k is a summary of the vector calculus presented in the
optional sections of this book. The first column shows that one func-
tion is a related to another by a kind of differentiation. The second
column states the fundamental theorem of calculus, which says that
if you integrate the derivative over the interior of a region, you get
some information about the original function at the boundary of
that region.710 Chapter 11 Electromagnetism