Page 218 - Engineering Electromagnetics, 8th Edition

200 ENGINEERING ELECTROMAGNETICS

of an inward normal to the surface of the page. If the velocity of water does not change
as we go up- or downstream and also shows no variation as we go across the river
(or even if it decreases in the same fashion toward either bank), then this component
is the only component present at the center of the stream, and the curl of the water
velocity has a direction into the page.
In Figure 7.14b, the streamlines of the magnetic field intensity about an infinitely
long filamentary conductor are shown. The curl meter placed in this field of curved
lines shows that a larger number of blades have a clockwise force exerted on them
but that this force is in general smaller than the counterclockwise force exerted on
the smaller number of blades closer to the wire. It seems possible that if the curvature
of the streamlines is correct and also if the variation of the field strength is just right,
the net torque on the paddle wheel may be zero. Actually, the paddle wheel does not
rotate in this case, for since H = (I/2πρ)a φ , we may substitute into (25) obtaining
1 ∂(ρH φ )
∂H φ
curl H =− a ρ + a z = 0
∂z ρ ∂ρ

EXAMPLE 7.2
As an example of the evaluation of curl H from the definition and of the evaluation of
2
another line integral, suppose that H = 0.2z a x for z > 0, and H = 0 elsewhere, as
shown in Figure 7.15. Calculate H · dL about a square path with side d, centered
at (0, 0, z 1 )inthe y = 0 plane where z 1 > d/2.

Figure 7.15 A square path of side d with its center on the
z axis at z = z 1 is used to evaluate H · dL and find curl H.