Page 217 - Engineering Electromagnetics, 8th Edition

CHAPTER 7 The Steady Magnetic Field 199

Figure 7.14 (a) The curl meter shows a component of the curl of the water velocity
into the page. (b) The curl of the magnetic field intensity about an infinitely long filament
is shown.

The circulation of H,or H · dL,is obtained by multiplying the component
of H parallel to the specified closed path at each point along it by the differential
path length and summing the results as the differential lengths approach zero and as
their number becomes infinite. We do not require a vanishingly small path. Amp`ere’s
circuital law tells us that if H does possess circulation about a given path, then current
passes through this path. In electrostatics we see that the circulation of E is zero about
every path, a direct consequence of the fact that zero work is required to carry a charge
around a closed path.
We may describe curl as circulation per unit area. The closed path is vanishingly
small, and curl is defined at a point. The curl of E must be zero, for the circulation
is zero. The curl of H is not zero, however; the circulation of H per unit area is the
current density by Amp`ere’s circuital law [or (18), (19), and (20)].
5
Skilling suggests the use of a very small paddle wheel as a “curl meter.” Our
vector quantity, then, must be thought of as capable of applying a force to each blade
of the paddle wheel, the force being proportional to the component of the field normal
to the surface of that blade. To test a field for curl, we dip our paddle wheel into the
field, with the axis of the paddle wheel lined up with the direction of the component of
curl desired, and note the action of the field on the paddle. No rotation means no curl;
larger angular velocities mean greater values of the curl; a reversal in the direction of
spin means a reversal in the sign of the curl. To find the direction of the vector curl and
not merely to establish the presence of any particular component, we should place
our paddle wheel in the field and hunt around for the orientation which produces the
greatest torque. The direction of the curl is then along the axis of the paddle wheel,
as given by the right-hand rule.
As an example, consider the flow of water in a river. Figure 7.14a shows the
longitudinal section of a wide river taken at the middle of the river. The water velocity
is zero at the bottom and increases linearly as the surface is approached. A paddle
wheel placed in the position shown, with its axis perpendicular to the paper, will turn
in a clockwise direction, showing the presence of a component of curl in the direction

5 See the References at the end of the chapter.