Page 434 - Electromagnetics

N
1 ˜

+ (dl · E)∇ G. (6.35)
jω ˜µ
n=1
na + nb
This is known as the vector Huygens principle after the Dutch physicist C. Huygens, who
formulated his “secondary source concept” to explain the propagation of light. According
e
to his idea, published in Trait´ de la lumi`ere in 1690, points on a propagating wavefront
are secondary sources of spherical waves that add together in just the right way to produce
the field on any successive wavefront. We can interpret (6.34) and (6.35) in much the
same way. The field at each point within V , where there are no sources, can be imagined
to arise from spherical waves emanated from every point on the surface bounding V . The
amplitudes of these waves are determined by the values of the fields on the boundaries.
Thus, we may consider the boundary fields to be equivalent to secondary sources of the
fields within V . We will expand on this concept below by introducing the concept of
equivalence and identifying the specific form of the secondary sources.

6.3.2 The Franz formula
The vector Huygens principle as derived above requires secondary sources for the fields
within V that involve both the tangential and normal components of the fields on the
bounding surface. Since only tangential components are required to guarantee uniqueness
˜
˜
within V , we seek an expression involving only ˆ n × H and ˆ n × E. Physically, the normal
component of the field is equivalent to a secondary charge source on the surface while
the tangential component is equivalent to a secondary current source. Since charge and
current are related by the continuity equation, specification of the normal component is
superfluous.
To derive a version of the vector Huygens principle that omits the normal fields we
take the curl of (6.35) to get

N N
˜ ˜ ˜
∇× H(r,ω) = ∇× (ˆ n × H) ×∇ GdS + ∇× (ˆ n · H)∇ G dS +
n=1 S n n=1 S n
N N
c ˜ ˜
1
+ ∇× jω˜ (ˆ n × E)GdS + ∇× (dl · E)∇ G dS . (6.36)
jω ˜µ
n=1 S n n=1
na + nb

Now, using ∇ G =−∇G and employing the vector identity (B.43) we can show that

∇× f (r )∇ G(r|r ) =− f (r ) ∇× ∇G(r|r ) + ∇G(r|r ) ×∇ f (r ) = 0,
since ∇× ∇G = 0 and ∇ f (r ) = 0. This implies that the second and fourth terms of

(6.36) are zero. The first term can be modified using
˜
˜
˜

∇× ˆ n × H(r ) G(r|r ) = G(r|r )∇× ˆ n × H(r ) − ˆ n × H(r ) ×∇G(r|r )
˜
= ˆ n × H(r ) ×∇ G(r|r ),
giving
N N
˜ ˜ c ˜
∇× H(r,ω) = ∇× ∇× (ˆ n × H)G dS + ∇× jω˜ (ˆ n × E)GdS .
n=1 S n n=1 S n
© 2001 by CRC Press LLC