Page 231 - Engineering Electromagnetics, 8th Edition

CHAPTER 7 The Steady Magnetic Field 213

agree not to select a path that crosses this plane. Therefore, we cannot encircle I, and
a single-valued potential is possible. The result is seen to be

I
V m =− φ (−π< φ < π)
2π
and
I π
V mP =− φ =
8 4
The scalar magnetic potential is evidently the quantity whose equipotential sur-
faces will form curvilinear squares with the streamlines of H in Figure 7.4. This is
one more facet of the analogy between electric and magnetic fields about which we
will have more to say in the next chapter.
Letustemporarilyleavethescalarmagneticpotentialnowandinvestigateavector
magnetic potential. This vector field is one which is extremely useful in studying
radiation from antennas (as we will find in Chapter 14) as well as radiation leakage
from transmission lines, waveguides, and microwave ovens. The vector magnetic
potential may be used in regions where the current density is zero or nonzero, and we
shall also be able to extend it to the time-varying case later.
Our choice of a vector magnetic potential is indicated by noting that

∇ · B = 0
Next, a vector identity that we proved in Section 7.4 shows that the divergence of the
curl of any vector field is zero. Therefore, we select
B =∇ × A (46)

where A signifies a vector magnetic potential, and we automatically satisfy the con-
dition that the magnetic flux density shall have zero divergence. The H field is
1
H = ∇× A
µ 0
and
1
∇× H = J = ∇× ∇× A
µ 0
The curl of the curl of a vector field is not zero and is given by a fairly complicated
9
expression, which we need not know now in general form. In specific cases for which
the form of A is known, the curl operation may be applied twice to determine the
current density.

9 2 2 2
∇× ∇× A ≡∇(∇ · A) −∇ A. In rectangular coordinates, it may be shown that ∇ A ≡∇ A x a x +
2
2
2
∇ A y a y +∇ A z a z .In other coordinate systems, ∇ A may be found by evaluating the second-order
2
partial derivatives in ∇ A =∇(∇ · A) −∇ × ∇ × A.