Page 137 - Maxwell House

POYNTING's THEOREM 117

Table 1.5 can help to check that every term in (3.7) has the same unit dimension W m . The
3
⁄
assumption that the dielectric and magnetic constants are time-independent is not entirely
correct because of the polarization and magnetization effects in materials discussed in Chapter
2. However, all such transient effects usually require some attention but short living and thus
can be neglected in most cases.

The integral form of energy conservation law in the space-time domain. Eventually, the total
power balance can be calculated by integrating both sides in (3.7) over the stationary volume
V bounded by a closed surface A

2
⁄
− ∫ ∘ = ∫ + ∫ ( ∘ + ∘ ) 2 + ∫ ∘ ( x ) [W] (3.8)

Applying the divergent theorem (A.60) from Appendix to the last integral on the right-hand
side, we get

2
− ∫ ∘ = ∫ + ∫ ( ∘ + ∘ ) 2 + ∯ x ∘ (3.9)
⁄

The equity (3.9) expressing electromagnetic energy conservation law was introduced in 1884
by English physicist John Henry Poynting and is known as the integral form of Poynting’s
theorem. If so, the expression (3.7) can be treated as the differential form of Poynting’s theorem
or the continuity equation for power density.

Another probably more explicit physically and not so formal way to Poynting’s theorem is to
define the energy of EM fields straight from Lorentz’s force equation instead of formal
manipulation of (3.2). Indeed, we know that the tiny portion of mechanical work or energy
d required to move a charge particle Δ at the distance in E- and H-fields is equal (see

(1.11)) to d = ∘ and = , where is the time of movement. Then d =
Δ ( + x ) ∘ = (Δ) ∘ = ∘ . The 2 nd M ax well’s e quat ion says

that x − ⁄ = . Therefore, ⁄ = ( x − ⁄) ∘ . The vector identity

st
∘ ( x ) = ( x ) ∘ − ( x ) ∘ and the 1 Maxwell’s equation directly leads to (3.7)
giving us the identical balance of power.
Each term in (3.9) needs some explanation to get its physical meaning.

3.1.2 Power Delivered by Excitation Currents

From the discussion in Section 1.7 of Chapter 1, we know that the integral on the left-hand side of
(3.9) describes the power given to the electromagnetic field by sources

() = − ∫ ∘ [W] (3.10)

Here the product ∘ < 0 and > 0 due to the fact that the current charges donating
their energy to the fields slow down as they move against the electric force.