Page 45 - Maxwell House

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICS 25

∯ ∘ = (1.26)

introducing the vector of electric displacement strength or electric flux density
= [C/m ] (1.27)
2
0
The equity (1.27) is called the constitutive relation and was put in Table 1.7 as a relationship
supporting Maxwell’s equation. In order to simplify the notation in (1.27) and the following
equations, the records of time and coordinate dependence are suggested but omitted. From time
to time, we come back to full notation to avoid confusion. The numerical quantity of is
0
called sometimes the absolute dielectric constant of vacuum.

In a vacuum, the only difference between vectors E and D is the unit dimension dictated by the
chosen SI units. 3 Maxwell’s equation in the form of (1.26) describes the integral effect of all
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electric fields existing in some volume V bounded by a surface A. Practically, that is not enough.
We need more information: namely how electromagnetic fields are dispersed inside the volume.
For cell phone customer it is not sufficient to know that the nearest communication tower in
order and radiates electromagnetic energy. We must be sure that this energy is enough at the
client spot to receive signal. Therefore, we need to switch from integral to deferential or to
point-to-point field form of description. To illustrate how to do it let us assume that the
displacement vector has small z-component only or = ∆ and = = ∆∆
0
0
0

and in (1.26). Then ∘ = ∆ ∆∆ since is the vector or unit length and ∘ = 1. It

0
0
0
will be perfectly valid to represent the product ∘ as
∘ = ∆ ∆∆∆ = (1.28)
∆
Here is the volume of infinitesimal parallelepiped. Eventually, applying the same transform
to all components of displacement vector we obtain

∘ = � + + � (1.29)

Looking back at Table 1.6 of mathematical operators we have
∘ = ∘ (1.30)

One more step is to transform the right-hand side of (1.26) using the association (1.7) of the
charge with its volume density

= ∫

Substituting this equity and (1.30) into (1.26) and putting all terms together we have
∫ ( ∘ − ) = 0 (1.31)

Now, we can say the “magic” words repeating them with some variations many times later:
since the volume V in (1.31) is arbitrary and this equity must hold for all of them, the integrand
must be equal to zero that makes this equality right in general. Therefore,

∘ = (1.32)

The equity (1.32) is the differential form of 3 Maxwell’s equation included in Table 1.7.
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