Page 136 - Maxwell House

116 Chapter 3

will help us to identify each term in the balance of energy provided by Poynting’s theorem. Be
fearless, our story will be not so mazily and bloody.

The differential form of energy conservation law in the space-time domain. Let us start
associating this form with the first two Maxwell’s differential equations (see Table 1.7 in
Chapter 1) and omitting, for the sake of simplicity, the magnetic sources that have not been
detected yet

∘ x = + +

+ � � (3.2)

− x =
∘

Note that the term (check (1.76)) introduced in Section 1.7.1 of Chapter 1 was added to

take into consideration the sources of EM fields while was replaced according to the

constitutive relation (2.28) from Chapter 2. Taking scalar product or dotting the top equation in
(3.2) with and bottom one with , as depicted in (3.2), we get summing all the products

∘ x − ∘ x = ∘ + ∘ + ∘ + ∘ (3.3)

The vector identity (A.35) in Appendix, ∘ x − ∘ x = − ∘ ( x ), lets transform
(3.3) as

− ∘ ( x ) = ∘ + ∘ + ∘ + ∘ (3.4)

Note that vector identity we have applied is the vector analog of the well-known product rule
′
′
′
′
that can be written as ∙ ℎ − ℎ ∙ = − ( ∙ ℎ) . Formally replace → , → x , ℎ →
′
, ℎ → x and you get it. Substituting the constitutive relations = and =

in (3.4), then regrouping the terms and assuming for simplicity that both dielectric and
0
magnetic constants are independent on time we have
(3.5)
− ∘ = ∘ + ∘ + ∘ + ∘ ( x )
0

Bearing in mind that
∘ = 2
1 (∘) 1 (∘)
∘ = = � (3.6)

2 2
1 (∘) 1 (∘)
∘ = =
0
0
2 2
we finally can write (3.5) as the balance of power per volume unit
2 1 (∘) 1 (∘) 3 (3.7)
⁄
− ∘ = + + + ∘ ( x ) [W m ]
2 2