Page 214 - Engineering Electromagnetics, 8th Edition

196 ENGINEERING ELECTROMAGNETICS

Figure 7.13 An incremental closed path in
rectangular coordinates is selected for the
application of Amp ` ere’s circuital law to determine
the spatial rate of change of H.

distance x/2 from the center to the midpoint of side 1–2:

. ∂ H y 1
H y,1−2 = H y0 + x
∂x 2
Thus

. 1 ∂ H y
(H · L) 1−2 = H y0 + x y
2 ∂x
Along the next section of the path we have

. . 1 ∂ H x
(H · L) 2−3 = H x,2−3 (− x) =− H x0 + y x
2 ∂y
Continuing for the remaining two segments and adding the results,

. ∂ H y ∂ H x
H · dL = − x y
∂x ∂y
By Amp`ere’s circuital law, this result must be equal to the current enclosed by the
path, or the current crossing any surface bounded by the path. If we assume a general
.
current density J, the enclosed current is then I = J z x y, and

. ∂ H y ∂ H x .
H · dL = − x y = J z x y
∂x ∂y
or

H · dL . ∂ H y ∂ H x .
= − = J z
x y ∂x ∂y
As we cause the closed path to shrink, the preceding expression becomes more nearly
exact, and in the limit we have the equality
H · dL ∂ H y ∂ H x

lim = − = J z (18)
x, y→0 x y ∂x ∂y