Page 219 - Engineering Electromagnetics, 8th Edition

CHAPTER 7 The Steady Magnetic Field 201

Solution. We evaluate the line integral of H along the four segments, beginning at
the top:

2 2
1
1
H · dL = 0.2 z 1 + d d + 0 − 0.2 z 1 − d d + 0

2
2
= 0.4z 1 d 2
In the limit as the area approaches zero, we find
H · dL 0.4z 1 d 2

(∇× H) y = lim = lim = 0.4z 1
d→0 d 2 d→0 d 2
The other components are zero, so ∇× H = 0.4z 1 a y .
To evaluate the curl without trying to illustrate the definition or the evaluation of
a line integral, we simply take the partial derivative indicated by (23):

a x a y a z

∂ ∂ ∂ ∂
2
∂x ∂z ∂z
∇× H = ∂y = (0.2z )a y = 0.4za y
0.2z 0 0
2

which checks with the preceding result when z = z 1 .
Returning now to complete our original examination of the application of
Amp`ere’s circuital law to a differential-sized path, we may combine (18)–(20), (22),
and (24),

∂H z ∂H y ∂H x ∂H z
curl H =∇ × H = − a x + − a y
∂y ∂z ∂z ∂x

∂H y ∂H x
+ − a z = J (27)
∂x ∂y
and write the point form of Amp` ere’s circuital law,
∇× H = J (28)

This is the second of Maxwell’s four equations as they apply to non-time-varying
conditions. We may also write the third of these equations at this time; it is the point
form of E · dL = 0, or

∇× E = 0 (29)

The fourth equation appears in Section 7.5.