Page 403 - Engineering Electromagnetics, 8th Edition

CHAPTER 11 The Uniform Plane Wave 385

The two time derivatives in (67) can be rearranged as follows:
∂E ∂ 1
E · = D · E (68a)
∂t ∂t 2
and
∂H ∂ 1
µH · = B · H (68b)
∂t ∂t 2
With these, Eq. (67) becomes

∂ 1 ∂ 1
−∇ · (E × H) = J · E + D · E + B · H (69)
∂t 2 ∂t 2
Finally, we integrate (69) throughout a volume:

1
1
∂ ∂
− ∇ · (E × H) dv = J · E dv + D · E dv + B · H dv
vol vol vol ∂t 2 vol ∂t 2
The divergence theorem is then applied to the left-hand side, thus converting the
volume integral there into an integral over the surface that encloses the volume. On
the right-hand side, the operations of spatial integration and time differentiation are
interchanged. The final result is
d 1 d 1

− (E × H) · dS = J · E dν + dt D · E dν + dt B · H dν (70)
area vol vol 2 vol 2
Equation (70) is known as Poynting’s theorem. On the right-hand side, the first
integral is the total (but instantaneous) ohmic power dissipated within the volume. The
second integral is the total energy stored in the electric field, and the third integral is
3
the stored energy in the magnetic field. Since time derivatives are taken of the second
and third integrals, those results give the time rates of increase of energy stored within
the volume, or the instantaneous power going to increase the stored energy. The sum
of the expressions on the right must therefore be the total power flowing into this
volume, and so the total power flowing out of the volume is

(E × H) · dS W (71)
area
wheretheintegralisovertheclosedsurfacesurroundingthevolume.Thecrossproduct
E × H is known as the Poynting vector, S,

S = E × H W/m 2 (72)

which is interpreted as an instantaneous power density, measured in watts per square
2
meter (W/m ). The direction of the vector S indicates the direction of the instantaneous

3 This is the expression for magnetic field energy that we have been anticipating since Chapter 8.