Page 135 - Maxwell House

POYNTING's THEOREM 115

Introduction

In Section 1.2 of Chapter 1, we already discussed the ultimate relationship between symmetries
of nature and conservation laws, most fundamental principles of physics, and stated that
Maxwell’s equations possess all the required symmetries and, therefore, can be derived from
conservation laws. We preferred not follow this “ultra-triathlon” mathematical path and took
“stress-free” axiomatic way to “get” Maxwell’s equations. Evidently, if all in all we have done
in Chapter 1 is correct, the electromagnetic field energy and power conservation laws must be
not only the part of Maxwell’s equations but must be and will be derivable from them. It will
not be an exaggeration to say that Poynting’s theorem proving the conservation of energy for
electromagnetic fields is the electrodynamics groundwork. By today, any result in macroscopic
electrodynamics that is in contrary to this theorem should be treated as an error or highly
doubtful.
Warning. Don’t expect to grasp the following material wholly on the first pass. Please read it
one more time sitting with a pad of paper, reviewing the main ideas and writing in words what
the equations say in symbols.

3.1 ELECTROMAGNETIC FIELD CONSERVATION LAWS

3.1.1 Conservation of Energy in Space-Time Domain

There are two forms of the energy conservation law in electrodynamics formulated in power
terms: differential or the balance of power per macroscopic volume unit and integral letting
estimate the total power accumulated in some finite or infinite domain. Since the power is the
rate of doing work while the energy measures the capability of a system to do work, the critical
task is to verify and understand the energy conservation laws likewise in any other branch of
physics; energy can neither be created nor destroyed; rather, it transforms from one form to
another. Energy and power are measured in Joule and Watt = Joule/second, respectively. It is
worthwhile to point out that in terms of derived units (see Table 1.5) several products connected
to electromagnetic vectors can produce the quantities linked to energy and power:

2
3
⁄
∙ → [(I m ) ∙ (V m) = W m ]
⁄
⁄
∙ → [(I ∙ s m ) ∙ (V m) = W ∙ s m = J m ]� (3.1)
2
3
3
⁄
⁄
⁄
⁄
3
3
2
⁄
⁄
⁄
∙ → [(V ∙ s m ) ∙ (I m) = W ∙ s m = J m ]
⁄
In other words, we can expect that the product of the volume electric current density and E-
field strength defines the volume power density, while the products ∙ and ∙ should
define the energy accumulated by the electric and magnetic fields in unit volume. Furthermore,
1
it is a hint that EM fields should accumulate and carry energy. This Ariadne’ Ariadne’s thread
1 Greek myth. Ariadne gave Theseus a ball of red thread, and he unrolled it as he penetrated the labyrinth.
There Theseus found the Minotaur deep in the recesses of the labyrinth, killed it with his sword, and
followed the thread back to the entrance.