Page 215 - Engineering Electromagnetics, 8th Edition

CHAPTER 7 The Steady Magnetic Field 197

After beginning with Amp`ere’s circuital law equating the closed line integral of
H to the current enclosed, we have now arrived at a relationship involving the closed
line integral of H per unit area enclosed and the current per unit area enclosed, or
current density. We performed a similar analysis in passing from the integral form of
Gauss’s law, involving flux through a closed surface and charge enclosed, to the point
form, relating flux through a closed surface per unit volume enclosed and charge per
unit volume enclosed, or volume charge density. In each case a limit is necessary to
produce an equality.
If we choose closed paths that are oriented perpendicularly to each of the re-
maining two coordinate axes, analogous processes lead to expressions for the x and
y components of the current density,

H · dL ∂ H z ∂ H y

lim = − = J x (19)
y, z→0 y z ∂y ∂z
and

H · dL ∂ H x ∂ H z

lim = − = J y (20)
z, x→0 z x ∂z ∂x
Comparing (18)–(20), we see that a component of the current density is given by
the limit of the quotient of the closed line integral of H about a small path in a plane
normal to that component and of the area enclosed as the path shrinks to zero. This
limit has its counterpart in other fields of science and long ago received the name of
curl. The curl of any vector is a vector, and any component of the curl is given by
the limit of the quotient of the closed line integral of the vector about a small path in
a plane normal to that component desired and the area enclosed, as the path shrinks
to zero. It should be noted that this definition of curl does not refer specifically to a
particular coordinate system. The mathematical form of the definition is

H · dL

(curl H) N = lim (21)
S N →0 S N
where S N is the planar area enclosed by the closed line integral. The N subscript
indicates that the component of the curl is that component which is normal to the
surface enclosed by the closed path. It may represent any component in any coordinate
system.
In rectangular coordinates, the definition (21) shows that the x, y, and z compo-
nents of the curl H are given by (18)–(20), and therefore

∂ H z ∂ H y ∂ H x ∂ H z ∂ H y ∂ H x
curl H = − a x + − a y + − a z (22)
∂y ∂z ∂z ∂x ∂x ∂y