Simple Nature - Light and Matter

i/A cyclic permutation of x,

y, andz.

j/Example 17.

i.e.,

F=axxˆ+byˆx+cxˆ+dxyˆ+eyyˆ+fˆy.

The only terms whose curls we haven’t yet explicitly computed are
thea,e, andfterms, and their curls turn out to be zero (homework
problem 50). Only thebanddterms have nonvanishing curls. The
curl of this field is

curlF= curl (byxˆ) + curl (dxyˆ)

=bcurl (yxˆ) +dcurl (xyˆ) [scaling]

=b(−ˆz) +d(ˆz) [found previously]

= (d−b)ˆz.

Butanyfield in thex−yplane can be approximated with this
type of field, as long as we only need to get a good approximation
within a small region. The infinitesimal Amp`erian surface occurring
in the definition of the curl is tiny enough to fit in a pretty small re-
gion, so we can get away with this here. Thedandbcoefficients can
then be associated with the partial derivatives∂Fy/∂xand∂Fx/∂y.
We therefore have

curlF=

(

∂Fy

∂x

−

∂Fx

∂y

)

ˆz

for any field in thex−yplane. In three dimensions, we just need to

generate two more equations like this by doing a cyclic permutation

of the variablesx,y, andz:

(curlF)x=

∂Fz

∂y

−

∂Fy

∂z

(curlF)y=

∂Fx

∂z

−

∂Fz

∂x

(curlF)z=

∂Fy

∂x

−

∂Fx

∂y

A sine wave example 17

.Find the curl of the following electric field

E= (sinx)yˆ,

and interpret the result.

.The only nonvanishing partial derivative occurring in this curl is

(curlE)z=

∂Ey

∂x

= cosx,

Section 11.4 Ampere’s law in differential form (optional)` 709