Curl
Figure 1
|
The curl is a vector. It is a measure of the circulation of a vector field. The component of
the curl of the vector
H
in the
z
direction normal to the surface
is defined as
| (1) |
where 
, is the area of the rectangular
loop lying in the
xy
plane shown in Figure 1.
Since the distances are small, when
y
changes to
,
the value of the
x
component of
H
changes to
.
Integrating
H
around the rectangular loop shown in Figure 1 in the counter-clockwise direction,
| (2) |
Dividing Equation 2 by
gives the
z
component of the curl of
H .
| (3) |
The x and y components of the curl are found in a similar manner.
| (4) |
The curl can also be expressed as a determinant.
| (5) |
Equation 6 is the integral form of Faraday's law.
| (6) |
Replacing H by E in Equation 1. Plug Equation 6 into Equation 1.
Then consider a surface to be divided
into many small surfaces. Apply Equation 1 to each of the small surfaces and
sum over all the small surfaces. This results in the following
differential form for Faraday's law,
| (7) |
Also, since
| (8) |
It follows that
| (9) |
Equation 9 is the differential form of Ampere's law.
