Page 139 - Maxwell House

POYNTING's THEOREM 119

Assume that our instruments are detected that the energy stored inside V growths for some

reasons, i.e. in (3.15) ⁄ > 0. It cannot be done by field sources that are not presented

within V. Therefore, the law of conservation dictates and our common sense tells us that some
unknown to us field sources should be located outside V and deliver this extra energy through
the surface of area A. This brings to play the surface integral from (3.15) describing the power
influx

() = − ∯ (, ) ∘ (3.16)
Σ

Explicitly, the negativity of the cross product x in (3.15) and positivity of () means that
Σ
the energy flux (, ) 2 P in (3.16) is responsible for energy increase in volume V. In other words,
such flux can be treated as the additional source of electromagnetic fields in volume V
equivalent to some internal source current. The vector of field’s power flux density (, )
called Poynting’s vector is defined as the cross product
3

(, ) = x [(V/m) (A/m) = J/s = W/m ] (3.17)
2
Evidently, this vector is numerically equal to the at a point flow of power inward (negative
flow) or outward (positive flow) per unit surface. It is easy to check that the decline in the stored
energy when d () d < 0 must be accompanied by
⁄

the positive flax of power () that means some energy
Σ
leaves the region V.
If so, the question is raised immediately. How could the
energy fall away and in what form? Now we know the
answer – the EM energy is capable of coming and
leaving in the form of EM waves carrying energy
through space without any additional support! This
marvel was predicted theoretically by Maxwell’s in 1865
and experimentally verified by Heinrich Hertz only Figure 3.1.2 Normal
twelve years later in 1887. If so, the magnitude of the component of Poynting’s
Poynting vector (, ) = x in (3.17) can be vector
interpreted as the instantaneous power flow per unit area surface and its direction as the
trajectory of the electromagnetic wave propagation.

Notice that the definition (3.16) for the net power flow throughout the whole closed area
Σ
includes only the normal to the surface A component of Poynting’s vector as shown in Figure
3.1.2. Therefore, the net power flow depends only on the tangential to the surface A components
of electric and magnetic field and (3.16) can be written as

2 The product of ∘ < 0 since the inflow vector S and vector in the direction of outside normal
are antiparallel.
3 The symbol S for the Poynting’s vector appears to come from the German, “strahlvektor” meaning
radiation vector.