Page 43 - Maxwell House

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICS 23

2
= ∯ () ∘ = () ∘ ∯ = () ∘ 4
0
0 � (1.24)

() = 0
4 0 2
Here is the unit vector pointing radially from the charge. The well-known equity (1.24)
0
describing the electric field around a static point-like charge and is usually derived from
Coulomb’s law. Putting the sensor #1 next to the charge Q we can find the exerted electrical
force from (1.11) as ( = 0)
Δ (1.25)
= Δ () =
2 0
4 0
which is Coulomb’s law. Figure
7
1.6.6a demonstrates the reality of
electrical fields pushing up girl’s hair
in full agreement with the vector
equation (1.25) and Figure 1.6.6b .
8
Do the charges store electric energy? a) b)
In fact, they do not store the energy
at all. Instead, the energy is stored in Figure 1.6.6 a) A girl’s hair after touching a
the electric field surrounding the charged sphere, b) Vector of the electric field
particle. generated by positively charged sphere

1.6.5 Is The Inverse-Square Relation Imperative?

Gauss’s law (1.23) is one of the fundamental theorems in electrodynamics, and we proved that
the purely empirical Coulomb’s law was followed from it. In particular, Gauss’s law led us to
the exact inverse-square relation between electrical field and distance in (1.24) and (1.25). Note
that Newtonian gravitation law follows the same inverse-square relationship. Is it the law of
nature or something occasional? The best validation test [8] made in 1983 estimated that the
deviation from Coulomb’s law in the form −2+δ is tremendously small and does not exceed
δ ≤ (2.7 ± 3.1) ∙ 10 −16 . According to Purcell [9], the inverse-square law has been
experimentally verified over the range of 10 −13 mm < r < 100 000 km. Why do physicists pay
such great attention to the accuracy of the inverse-square law in Coulomb’s and Newton’s
gravitation law? The answer is quite dramatic for humans: if these forces should not be tuned
correctly, our universe must be very different. Some scientists formulated a very curious
anthropic principal [10] “ … the universe is it is because if it were different we would not be
here to observe it.” It means that variation in inverse-square laws can lead, for example, to
catastrophic implications such as the existence of the different kind of universes with no
opportunity to produce humans.
1.6.6 How Much Is One Coulomb (C)?

Quite powerful! Suppose two point-like charges of 1C each are located at a distance 1 km =
1000m. The force value can be calculated using (1.1) and the numerical constant

7 Public Domain Image, source: https://www.flickr.com/photos/57167312@N02/5270904732. This
work has been identified as being free of known restrictions under copyright law, including all related
and neighboring rights.
8 Public Domain Image, source: http://inspirehep.net/record/946729/files/CoulombsLaw.png