Simple Nature - Light and Matter

is very much as it was before, except that all the field vectors have

had one unit worth ofyˆadded to them. But what do we get if we

take the curl of−xyˆ+yˆ? The curl, like any god-fearing derivative

operation, has the additive property

curl (F+G) = curlF+ curlG,

so

curl (−xyˆ+yˆ) = curl (−xyˆ) + curl (yˆ).

But the second term is zero, so we get the same result as at the

origin.

A field that goes in a circle example 15

.What is the curl of the fieldxyˆ−yxˆ?

.Using the linearity of the curl, and recognizing each of the terms

as one whose curl we have already computed, we find that this

field’s curl is a constant 2ˆz. This agrees with the right-hand rule.

The field inside a long, straight wire example 16

.What is the magnetic fieldinsidea long, straight wire in which

the current density isj?

.Let the wire be along thezaxis, soj=jˆz. Ampere’s law gives`

curlB=

4 πk

c^2

jˆz.

In other words, we need a magnetic field whose curl is a constant.

We’ve encountered several fields with constant curls, but the only

one that has the same symmetry as the cylindrical wire isxyˆ−yxˆ,

so the answer must be this field or some constant multiplied by it,

B=b(xyˆ−yxˆ).

The curl of this field is 2bˆz, so

2 b=

4 πk

c^2

j,

and thus

B=

2 πk

c^2

j(xyˆ−yxˆ).

The curl in component form

Now consider the field

Fx=ax+by+c

Fy=dx+ey+f,

708 Chapter 11 Electromagnetism