Page 390 - Electromagnetics

We may thus decompose any vector into a sum of longitudinal and transverse parts. An
important consequence of Maxwell’s equations is that the transverse fields may be written
entirely in terms of the longitudinal fields and the sources. This holds in both the time
and frequency domains; we derive the decomposition in the frequency domain and leave
the derivation of the time-domain expressions as exercises. We begin by decomposing
the operators in Maxwell’s equations into longitudinal and transverse components. We
note that
∂
≡ ˆ u ·∇
∂u
and define a transverse del operator as
∂
∇ t ≡∇ − ˆ u .
∂u
Using these basic definitions, the identities listed in Appendix B may be derived. We
shall find it helpful to express the vector curl and Laplacian operations in terms of
their longitudinal and transverse components. Using (B.93) and (B.96) we find that the
transverse component of the curl is given by

(∇× A) t =−ˆ u × ˆ u × (∇× A)

∂A t
ˆ
=−ˆ u × ˆ u × (∇ t × A t ) − ˆ u × ˆ u × u × −∇ t A u . (5.98)
∂u
The first term in the right member is zero by property (B.91). Using (B.7) we can replace
the second term by

∂A t ∂A t
ˆ
−ˆ u ˆ u · u × −∇ t A u + (ˆ u · ˆ u) ˆ u × −∇ t A u .
∂u ∂u
The first of these terms is zero since

∂A t ∂A t
ˆ
ˆ u · u × −∇ t A u = −∇ t A u · (ˆ u × ˆ u) = 0,
∂u ∂u
hence

∂A t
(∇× A) t = ˆ u × −∇ t A u . (5.99)
∂u
The longitudinal part is then, by property (B.80), merely the difference between the curl
and its transverse part, or

ˆ u (ˆ u ·∇ × A) =∇ t × A t . (5.100)
A similar set of steps gives the transverse component of the Laplacian as

2

2 ∂ A t
(∇ A) t = ∇ t (∇ t · A t ) + 2 −∇ t ×∇ t × A t , (5.101)
∂u
and the longitudinal part as
2 2
ˆ u ˆ u ·∇ A = ˆ u∇ A u . (5.102)
Verification is left as an exercise.

© 2001 by CRC Press LLC