Simple Nature - Light and Matter

f/The field−yxˆ.

g/Example 14.

h/Example 15.

by evaluating the field at the midpoints of its sides,

x=s/ 2 y= 0 F= (s/2)yˆ s 1 ·F=s^2 / 2

x= 0 y=s/ 2 F= 0 s 2 ·F= 0

x=−s/ 2 y= 0 F=−(s/2)yˆ s 3 ·F=s^2 / 2

x= 0 y=−s/ 2 F= 0 s 4 ·F= 0,

which gives a circulation of s^2 , and a curl with a magnitude of
s^2 /area =s^2 /s^2 = 1. By the right-hand rule, the curl points out of
the page, i.e., along the positivezaxis, so we have

curlxyˆ=ˆz.

Now consider the field−yxˆ, shown in figure f. This is the same

as the previous field, but with your book rotated by 90 degrees about

thezaxis. Rotating the result of the first calculation,zˆ, about the

zaxis doesn’t change it, so the curl of this field is alsoˆz.

Scaling

When you’re taking an ordinary derivative, you have the rule

d

dx

[cf(x)] =c

d

dx

f(x).

In other words, multiplying a function by a constant results in a

derivative that is multiplied by that constant. The curl is a kind of

derivative operator, and the same is true for a curl.

Multiplying the field by− 1. example 14

.What is the curl of the field−xˆyat the origin?

.Using the scaling property just discussed, we can make this into

a curl that we’ve already calculated:

curl (−xyˆ) =−curl (xˆy)

=−ˆz

This is in agreement with the right-hand rule.

The curl is additive.

We have only calculated each field’s curl at the origin, but each

of these fields actually has the same curl everywhere. In example

14, for instance, it is obvious that the curl is constant along any

vertical line. But even if we move along thexaxis, there is still an

imbalance between the torques on the left and right sides of the curl-

meter. More formally, suppose we start from the origin and move

to the left by one unit. We find ourselves in a region where the field

Section 11.4 Ampere’s law in differential form (optional)` 707