Page 163 - Maxwell House

POYNTING's THEOREM 143

space ( () = () = 0). Then the balance of the in-phase (real or active) power for such
′′
′′

source follows from (3.49)
1 = − ∯ ℜ( x ) ∘ (3.71)
1
∗
2
Here is the sphere of large radius r centered at the
source location as shown in Figure 3.3.1. The equity
(3.71) tells us that the crossing the boundary surface
power (the term on the right-side (3.71)) is
independent on the radius of the sphere. On the other
hand, we know that the sphere surface = 4
2
increases proportional to the radius square. If so, the
integrant in (3.71) must vanish at the same rate that
Figure 3.3.1 Sphere with current
source inside keeps integral unaffected or

lim( |ℜ( x )|) = . (3.72)
2
∗
→∞
Therefore, the fields produced by sources within an unbounded region are unique as long as
the asymptotical condition (3.72) called the radiation condition is fulfilled. If you found that
your numerical or analytical solution does not satisfy (3.72), start from the beginning and look
for specific errors in your approach. Note that there are many different forms of the radiation
condition. We will not pursue this subject here.

It may seem surprising the supremacy of limit (3.72) that restricts the field behavior of field
sources including all kind of antennas: single monopole or dipole, any finite combination of
them, horns or dish antennas, radar or satellite antennas, etc.

3.3.2 Edge Boundary Conditions

Figure 3.3.2 Objects with sharp edges: a) Aircraft, b) Ship, c) Top of mountain

Another situation where Maxwell’s equations solutions may not be unique emerges if the
13
objects under consideration contain the profiles with sharp edges (see Figure 3.3.2 ). It can be

13 Public Domain Image, source: https://en.wikipedia.org/wiki/Lockheed_Martin_F-35_Lightning_II,
http://www.navy.mil/ah_online/zumwalt/, https://en.wikipedia.org/wiki/Arches_National_Park