Page 154 - Maxwell House

134 Chapter 3

According to (2.68) and (2.72)

∗
∗
′

∗
∗
∗
∗
′′

∘ = ∘ ( () ) = () ∘ = ()|| = ( () + ())||
0
0
0
0

∘ = ∘ ( ()) = () ∘ = ()|| = ( () − ())||
′′
∗
∗
′

∗

0
0
0
0

Therefore, the equation (3.46) can be rewritten as
1 ∗ ′′ ′′ ′ ′
− ( ) ∘ = [ ()|| + ()|| ] − [ ()|| − ()|| ]

0
0

2 2 2 (3.47)
1 ∗ 3
+ ∘ ( x ) [W m ]
⁄
2
′′
′′
′′
Here () = () + and () is defined by (2.71) and (2.73). Both are the imaginary

0
part of the relative complex dielectric and magnetic constant, respectively.
The integral form of energy conservation law in space-frequency domain. Understandably,
the total power balance can be calculated by integrating both sides in (3.47) over the stationary
volume V bounded by a closed surface A. Then applying the divergence theorem (A.60) from
Appendix to the last term in (3.47) we have
() = () + () + () (3.48)

Σ
Here
1 ∗
() = − ∫ ( ) ∘

2 ⎫
⎪
⎪

′′
′′

() = ∫ [ ()|| + ()|| ]/2 (3.49)
0
0

′

′
⁄
() = 2 ∫ [ ()|| + ()|| ] 4 ⎬

⎪
⎪
∗
() = ∯ ( x )/2 ∘ ⎭
Σ

The equity (3.48) expressing electromagnetic energy conservation is the integral form of
Poynting’s theorem for monochromatic fields. If so, (3.47) can be treated as the differential
form of this theorem or the continuity equation for power density.
The last equity in (3.49) brings to play the surface integral describing the power influx
() = ∯ (, ) ∘ (3.50)
Σ

Here the vector of field’s power flux density (, ) is called the complex Poynting’s vector,
defined as the cross product
(, ) = ( x )/2 (3.51)
∗
It is evidently equal to the at a point flow of time-average power inward (negative flow) or
outward (positive flow) per unit surface. Explicitly, the negative magnitude of the cross product