Page 140 - Maxwell House

120 Chapter 3

() = ∯ x (3.18)
Σ

Poynting’s vector is probably the most interesting component of energy conservation law
because behind it the quantum nature of EM fields. Loosely speaking, this vector is the
“macroscopic bridge” bonding classical Maxwell’s equations with quantum physics.

3.1.6 Velocity of EM Waves Energy Transportation

Poynting’s theorem as one of the key theorem of
electrodynamics predicts not only the effect of
electromagnetic energy transportation by EM waves
but allows to estimate the velocity of this transportation
customary called energy velocity of EM wave. Let us
refer to Figure 3.1.3 assuming that it takes some short
period of time ∆ for the incoming EM energy ∆ to

fill up the tiny volume ∆ = ∆ ∘ ∆. Then the balance
of power contained inside this volume (see (3.15)) can
be written as
Figure 3.1.3 Volume ∆ filling
up with energy flux ∆ ⁄ ∆ = − ∬ ∘ = ∘ ∆ (3.19)

∆
Taking into account that the volume is infinitesimal we obtain

∫ ∘ ∆ ∆
∆ ⁄ ∆ = ∆ = ∘ ∫ = ∘ ( ∆) (3.20)

∆ ∆ ∆ ∆
∆
The ratio can be interpreted as the velocity vector of the energy flux propagating in the
∆
direction ∆. Therefore,

⁄
∘ ( ∆) = ∘ ∆ or = [m/s] (3.21)
Note that we implicitly assumed that the energy flux through the side surface of ∆ is small
enough to be neglected and the medium of propagation is isotropic and non-dispersive. In this
case, the Poynting’s vector direction shows the path and velocity of the energy flux, i.e. ||.
The detailed discussion of this subject is out of this book theme. More fundamental analysis of
such issue the reader can find in [6].

3.1.7 Linear Momentum of EM Fields. Radiation Pressure and Solar Sailing

Feedback between Poynting’s theorem and conservation laws is much deeper than it seems at
first glance. Let us come back for a while to the EM wave-particle duality we have touched in
4
Chapter 1. It is well-known that each photon carries a certain portion of energy = ℎ =
2
traveling in vacuum at the ultimate speed of light and never stops. Meanwhile, it sounds creasy

4 To avoid the confusion, we temporally use the common in physics notification of energy and frequency.