Page 216 - Engineering Electromagnetics, 8th Edition

198 ENGINEERING ELECTROMAGNETICS

This result may be written in the form of a determinant,

a x a y
a z
∂ ∂ ∂

∂x ∂y ∂z
curl H = (23)

H x H y H z
and may also be written in terms of the vector operator,

curl H =∇ × H (24)
Equation (22) is the result of applying the definition (21) to the rectangular coordi-
nate system. We obtained the z component of this expression by evaluating Amp`ere’s
circuital law about an incremental path of sides x and y, and we could have ob-
tainedtheothertwocomponentsjustaseasilybychoosingtheappropriatepaths.Equa-
tion (23) is a neat method of storing the rectangular coordinate expression for curl; the
form is symmetrical and easily remembered. Equation (24) is even more concise and
leads to (22) upon applying the definitions of the cross product and vector operator.
The expressions for curl H in cylindrical and spherical coordinates are derived in
Appendix A by applying the definition (21). Although they may be written in determi-
nant form, as explained there, the determinants do not have one row of unit vectors on
top and one row of components on the bottom, and they are not easily memorized. For
this reason, the curl expansions in cylindrical and spherical coordinates that follow
here and appear inside the back cover are usually referred to whenever necessary.

1 ∂H z ∂H φ ∂H ρ ∂H z
∇× H = − a ρ + − a φ
ρ ∂φ ∂z ∂z ∂ρ
1 ∂(ρH φ ) 1 ∂H ρ
(25)
+ − a z (cylindrical)
ρ ∂ρ ρ ∂φ
1 ∂(H φ sin θ) ∂H θ 1 1 ∂H r ∂(rH φ )
∇× H = − a r + − a θ
r sin θ ∂θ ∂φ r sin θ ∂φ ∂r (26)
1 ∂(rH θ ) ∂H r

+ − a φ (spherical)
r ∂r ∂θ

Although we have described curl as a line integral per unit area, this does not
provideeveryonewithasatisfactoryphysicalpictureofthenatureofthecurloperation,
for the closed line integral itself requires physical interpretation. This integral was
first met in the electrostatic field, where we saw that E · dL = 0. Inasmuch as the
integral was zero, we did not belabor the physical picture. More recently we have
discussed the closed line integral of H, H · dL = I. Either of these closed line
integrals is also known by the name of circulation, a term borrowed from the field of
fluid dynamics.