Page 223 - Engineering Electromagnetics, 8th Edition

CHAPTER 7 The Steady Magnetic Field 205

Thus, the results check Stokes’ theorem, and we note in passing that a current of
22.2 A is flowing upward through this section of a spherical cap.

Next, let us see how easy it is to obtain Amp`ere’s circuital law from ∇× H = J.
We merely have to dot each side by dS, integrate each side over the same (open)
surface S, and apply Stokes’ theorem:

(∇× H) · dS = J · dS = H · dL
S S
The integral of the current density over the surface S is the total current I passing
through the surface, and therefore

H · dL = I
This short derivation shows clearly that the current I, described as being “en-
closed by the closed path,” is also the current passing through any of the infinite
number of surfaces that have the closed path as a perimeter.
Stokes’ theorem relates a surface integral to a closed line integral. It should
be recalled that the divergence theorem relates a volume integral to a closed surface
integral. Both theorems find their greatest use in general vector proofs. As an example,
let us find another expression for ∇ · ∇× A, where A represents any vector field. The
result must be a scalar (why?), and we may let this scalar be T ,or
∇ · ∇× A = T

Multiplying by dν and integrating throughout any volume ν,

(∇ · ∇× A) dν = Tdν
vol vol
we first apply the divergence theorem to the left side, obtaining

(∇× A) · dS = Tdν
S vol
The left side is the surface integral of the curl of A over the closed surface
surrounding the volume ν. Stokes’ theorem relates the surface integral of the curl of
A over the open surface enclosed by a given closed path. If we think of the path as
the opening of a laundry bag and the open surface as the surface of the bag itself, we
see that as we gradually approach a closed surface by pulling on the drawstrings, the
closed path becomes smaller and smaller and finally disappears as the surface becomes
closed. Hence, the application of Stokes’ theorem to a closed surface produces a zero
result, and we have

Tdν = 0
vol