Page 220 - Engineering Electromagnetics, 8th Edition

202 ENGINEERING ELECTROMAGNETICS

D7.4. (a)Evaluate the closed line integral of H about the rectangular path
P 1 (2, 3, 4) to P 2 (4, 3, 4) to P 3 (4, 3, 1) to P 4 (2, 3, 1) to P 1 ,given H = 3za x −
3
2x a z A/m. (b) Determine the quotient of the closed line integral and the area
enclosed by the path as an approximation to (∇× H) y .(c) Determine (∇× H) y
at the center of the area.
2
Ans. 354 A; 59 A/m ;57A/m 2

D7.5. Calculate the value of the vector current density: (a)in rectangular
2
2
coordinates at P A (2, 3, 4) if H = x za y − y xa z ;(b)incylindrical coordi-
2
◦ (cos 0.2φ)a ρ ;(c)in spherical coordinates at
nates at P B (1.5, 90 , 0.5) if H =
ρ
1
◦ ◦ a θ .
P C (2, 30 , 20 )if H =
sin θ
2
2
Ans. −16a x + 9a y + 16a z A/m ;0.055a z A/m ; a φ A/m 2
7.4 STOKES’THEOREM
Although Section 7.3 was devoted primarily to a discussion of the curl operation,
the contribution to the subject of magnetic fields should not be overlooked. From
Amp`ere’s circuital law we derived one of Maxwell’s equations, ∇× H = J. This
latter equation should be considered the point form of Amp`ere’s circuital law and
applies on a “per-unit-area” basis. In this section we shall again devote a major share
of the material to the mathematical theorem known as Stokes’ theorem, but in the
process we will show that we may obtain Amp`ere’s circuital law from ∇× H = J.
In other words, we are then prepared to obtain the integral form from the point form
or to obtain the point form from the integral form.
Consider the surface S of Figure 7.16, which is broken up into incremental
surfaces of area S.Ifwe apply the definition of the curl to one of these incremental
surfaces, then

H · dL S .
= (∇× H) N
S
where the N subscript again indicates the right-hand normal to the surface. The
subscript on dL S indicates that the closed path is the perimeter of an incremental
area S. This result may also be written

H · dL S .
= (∇× H) · a N
S
or

.
H · dL S = (∇× H) · a N S = (∇× H) · S
where a N is a unit vector in the direction of the right-hand normal to S.
Now let us determine this circulation for every S comprising S and sum the re-
sults. As we evaluate the closed line integral for each S, some cancellation will occur