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Lightning always fascinated men,with a mixture of fear and amazement for the colossal
natural forces that are released during the storms.

Since the most remote times the blue light of lightning and the din of thunders have been
associated , within the various popular cultures, to mysterious forces of divine origin
released against the man as to underline the power of abysmal discrepancy between the
divine and the human reality.

Ancient Greeks believed lightning was cast by Zeus against men for their serious offences
to divinity.

With the progress of the knowledge, men began understanding the physicalness of the
natural electric phenomena, and in particular,in more recent time, since the XVII century,they
observed electric phenomena with a speculative and not any longer superstitious attitude.

In particular Gilbert,the personal physician of Queen Elizabeth I, effected the first
studies on electrostatic phenomena, taking account of the electric property of some
materials, like, for example, glass, of attracting small pieces of various substances,
after they have been rubbed by a cloth or a fur, a phenomenon already observed for amber
(named "elektron" in Greek) in the ancient Greece.

This researches permitted him to establish that the electric charges acquired by two
electrified bodies by means of continuous rubbing, are always of two different kinds, one
positive and the other one negative.

Electrification phenomena, initially produced as hobbies,by using electrostatic machines
operating by continuous rubbing, were transformed little by little from parlour games into
scientific experiments, till the French physicist Coulomb, in the second half of XVIII
century, enunciated the law that is known with his name, relative to the attractive or
repulsive forces acting between two pointlike electric charges, respectively of opposite
sign or of the same sign, mathematically analogous to Newton's universal gravitation law.

Electrostatics received noticeable contributions from the researches made by Benjamin
Franklin and Michel Faraday ,who also invented, respectively, the lightning-conductor and
the electrostatic screen (the so-called Faraday's cage).

A noticeable impulse to developing the researches about electromagnetic phenomena
was given by the invention of Volta's battery (1800 ), by which were effected the
experiments on the electric currents and their effects (thermal, magnetic and chemical ).

These researches continued to progress till the formulation of the famous equations of the
electromagnetism by the Scottish physicist Clerk Maxwell (during the second half of XIX
century ),who furnished for the first time a theoretical and complete vision of all the
electric and magnetic phenomena.

The electric charge is one of the several characteristics that differentiate the
fundamental particle of matter, of which it may be considered as a particular
"flavor".

In Nature exist both neutral particles, that is particles with no electric charge and
particles with a positive or negative electric charge,that have an electric
"flavor" .

This particular attribute of matter produces all the electric, magnetic and
electromagnetic phenomena, and it is the source of the respective fields.

What does it mean field? To understand this concept we may consider a scalar field.

What is the meaning of scalar field ?

A simple example is provided by the dependence of temperature or of the barometric
pressure on the geographical coordinates of an assigned place.

In both cases we must consider scalar physical quantities , that are defined only by a
number expressing their measure relating to an unitary physical quantity.

Since temperature or pressure depend on the place we consider, we speak about thermal
field or barometric field, respectively.

If instead we consider a vectorial physical quantity, for example the gravity force acting
on a body placed on the ground at the sea level, we have to specify as vary the intensity
and the direction of the weight force in relation to the latitude and the longitude of the
place: at the poles the weight will be greater that the one measured at the Equator.

In the first case, indeed, since is zero the distance between the body and the Earth axis,
is zero also the centrifugal force produced by the Earth rotation, that produces a
lightening of the body; then the weight of the body is greater than the one measured in
the other case, when the lightening produced by the centrifugal force must be taken into
account, because the distance between the body and the Earth axis

is maximal.

Obviously, in intermediate places between the Equator and the poles, the gravity force
varies between both preceding extreme cases.

Therefore on the Earth surface the gravity force measured at the sea level, is a vectorial
field.

The attraction or repulsive forces between two pointlike electric charges are described
in classical physics by Coulomb's law,according to which the force is attractive if the
charge polarities are opposite and repulsive if they have the same sign, and it is
directly proportional to the product of the charges and inversely proportional with
respect to the square of the distance R that separates them.

For example, if two pointlike electric and opposite charges Q_{1}
and Q_{2},separated by a distance of 1 cm, are
attracting each other by the force with modulus F,other two
charges Q_{1}' and Q_{2}',
10 times greater than Q_{1} and Q_{2},
are acting each other with the attractive force F' having a
modulus 100 times greater times than F.

If instead,with an assigned product of the charges, the distance R
between them becomes double or triple, the values of the attraction forces become
respectively four or nine times smaller:

F = k Q_{1} Q_{2}/R^{2},where k is a proportionality constant
(the Coulomb constant).

Coulomb's law founds, as Newton's universal attraction law, on the classical remote action
principle,according to which an electric charge placed in any medium (oil,air,vacuum)
attracts or rejects instantly another charge directly, that is without any physical medium
that transmits the force between both charges.

This principle dominates all the prerelativistic physics, and implicates the istantaneous
action of the Coulomb force, compatible with assuming the transmission speed of the
interaction to be infinite.

It must be considered that, before the formulation of the relativity theory (1905) by
Albert Einstein, the physicists had realized that the model of the istantaneous and direct
remote action, couldn't be any

longer physically acceptable. Therefore they started to think about the electric field as
the physical intermediary of the interaction between two electric charges.

Therefore,an electric charge at rest produces an electric field in the space surrounding
it, and another electric charge doesn't attracted or rejected instantly, but with a delay
given by the ratio r/c, between the distance r between the charges and the light speed c
= 300000 km/s.

The other charge feels the attraction or the repulsion not directly by the first charge,
but exclusively by means of the electric field, that is the physical medium that permits
the propagation of the interaction, that therefore isn't considered istantaneous any
longer, but resulting from many elementary adjoining interactions, defined in every point
of the space and caused by the electric field produced by the first charge.

Obviously,and in a symmetrical mode, the phenomenon can be understood as the attractive
interaction between the charges, caused by the electric field generated by the second
charge and felt by the first one.

The intensity of the electric field produced by an electric charge at any point of the
space can be determined by measuring the force acting on a smalless electric charge, used
as a testing charge and placed in an assigned point, and dividing the intensity of the
force by the number that expresses in coulombs (C) the measure of the testing charge.

If, for example, has to be measured the intensity of the electric field generated by a
charge of Q coulomb in all the points at a distance of R meters from the charge,that for simplicity is considered to be
pointlike, it is necessary to place a positive testing charge q,
the smallest as possible, to avoid that the electric field produced by Q is modified sensitively by the electric field produced by q, and to measure with a sensitive dynamometer the intensity of the
Coulomb force F ( expressed in newtons) acting on q.

To get the intensity of the electric field in the considered points, the intensity of the
measured force has to be divided by the magnitude (in coulomb) of the testing charge.

Therefore the intensity and direction of the spherical symmetric electric field E produced by the charge Q , are
gotten by the formula

E =F/q = (KQq/R^{2})/q = KQ/R^{2}.

If, for example, in a point at a distance of 0,6 m from q,
the force F has the intensity of 15 mN ( thousandths of
newton ) and the testing charge q has the value of 5 mC ( 5
thousandths of coulomb ), the electric field has the intensity E
= 15.10^{-3}: 5.10-^{3} = 3 N/C = 3V/m, and it is a vector directed from Q toward q if q is
positive (repulsive force), and in the opposite direction (attractive force) if q is negative, and has the same intensity in all the points of a
sphere having a radius of 0,6 m and the center in Q.

When are considered the electric charges moving in vacuum, in the air, in a metal or in
an watery solution of an acid, of a base or of a salt (an electrolytic solution containing
positive and negative ions ), under the action of a constant electric field, it is more
simple to consider differences of electrostatic potential energy to calculate as vary the
kinetic energy and speed of ions.

This is a problem analogous to the one of a body moving on the ground, being subjected to
the gravity force.

For example,if we kick a football on a rising ground, little by little the speed will be
diminishing by means of the gravity force, whose component along the inclined plane is
directed in the opposite direction with respect to the speed, till the football is stopped
after having covered a distance that is the greater, the greater has been the speed
initially impressed.

The football is stopped when the difference of height between the the stopping point and
the throwing one will be such that the corresponding difference of gravitational potential
energy equates the initial kinetic energy

K = m V^{2}/2 impressed by the footballer, where m is the mass of the football and V.

This happens because the gravitational field, as the electric one, is a conservative
field, such that the work made by the force of gravity on a body moving along an inclined
plane, depends only on the gradient of the gravitational potential energy between the
throwing point and the stopping one, and it is calculated by multiplying the weight force P (expressed in newtons) by the height difference (H_{2}-H_{1} ) (expressed in meters).

Since the weight force P is gotten multiplying the mass m in kg by the gravity acceleration g
(about 9,8 m/s^{2} at the sea level and at a latitude of 45°), that represents
the Earth gravitational field, the negative work L (in
joules) made by the gravity force, that decelerates the football, is given by:

L =- P (H_{2}-H_{1}) = -m g (H_{2}-H_{1}
) = m g H_{1} - m g H_{2}, where the expression mgh is the
gravitational potential energy of the mass m placed at the
height H on the ground, and L
equalizes the difference between the initial gravitational potential energy and the final
one.

If,for example, we consider a ball having a mass of 0,7 kg, and by a kick we throw it
along an inclined plane to make it reach the altitude H_{2}
= 1, 5 m in correspondence of the stopping instant, taking into account that at the base
of the inclined plane H_{1} = 0, the gravitational
field makes the negative work L = - 0,7 x 9, 8 x 1, 5 = -10,
29 J and equates, in absolute value, the kinetic energy K = m V^{2}/2
impressed initially to the ball.

The expression gH is the gravitational potential energy per an
unitary mass,that is the so-called gravitational potential U = gh
, proportional to the height H.

Therefore the work L may also be calculated by multiplying
the mass m by the gravitational potential difference U_{1}-U_{2} = g ( H_{1}-H_{2} ),

hat is L = m (U_{1}-U_{2}).

Likewise, if we consider, for example, the constant electric field generated by a battery
whose poles are connected to a capacitor, which consists of two parallel metallic planes
placed in the air at the distance d little in comparison with
their dimensions, we get an uniform electric field,characterized by parallel force lines
and a constant intensity inside the volume delimited by the planes.

In this particular case,which is analogous to the one of the gravitational field in the
proximity of the Earth surface, the electrostatic potential in a point at the distance x from the plane connected to the negative pole,that is the
electrostatic potential energy per an unitary electric charge,is expressed by V = E x .

According to this convention it is assumed V_{1} = 0
on the negative plane and V_{2 }= E d on the positive
plane.

Considering that in this case the electrostatic potential difference V_{2}-V_{1}
between the planes equalizes the tension (expressed in Volts) of the battery, we calculate
that with a 12 V battery connected to the capacitor metallic planes placed at the distance
d = 1 mm = 0,001 m, the intensity of the uniform electric
field inside the capacitor is:

E = (V_{2}-V_{1}) /d =
12/0,001 = 12.000 V/m = 12.000 N/C, that corresponds to a force of 12.000 N (newtons)
acting on a body with an unitary electric charge (1 coulomb ).

For example, we suppose to introduce into this uniform electric field E,near the plane connected to the negative pole of the battery, a
smallest sphere made of an insulating material ( with a radius much smaller than the
distance d between the conductor planes),and having a
negative electric charge q of 10 mC (q
=-0,01 coulomb).

The sphere is moving under the action of a force F given by
the product of q for E, that is F = E q = 12.000 x 0, 01 = 120 N,
directed from the points at a smaller potential toward the ones at a greater potential.

The work L made by the electric field on the spherule, will
be numerically equal to the increase of its kinetic energy K
along the distance d between the conductor planes, and it is
gotten by multiplying q = 0,01 C
by the potential difference

V_{2}-V_{1} =
0 - 12 V = -12 V, that is L = q (V_{2}-V_{1})
=

= - 0, 01 x ( 0 - 12 ) = 0,12 J (joule) = K_{final} - K_{initial}
.

If the charge q is positive and is placed initially near the
the positive conductor plane,it is moving toward the negative conductor plane,that is from
the points at a greater potential toward the ones at a smaller potential, and the work
made L is given by:

L = q ( V_{2}-V_{1}) = 0,01 x ( 12 -0 ) =
0,12 J = K_{final} - K_{initial }.

The value of the electrostatic potential is not definite in an univocal mode, because it
contains always, as the gravitational one, an arbitrary constant, that is eliminated when
the potential difference between two points is considered.

The potential difference makes it possible to calculate, by considerations founded on the
energy conservation principle, the variations of the kinetic energy, and then of the speed
of a charged body that is moving under the action of the electric forces produced by the
field.

The fundamental physical difference between the gravitational field and the electric one
consists in the fact that, while the acceleration a acquired
by a charged body subjected to an electric field, varies with the inverse proportionality
law with respect to the mass m of the body, with an assigned
electric charge q, that it a
becomes one half or one third, if the mass m respectively is
becoming double or triple, instead in the case of the gravitational field g, for the Galileo principle, all the bodies, independently from
their mass, are moving with the same acceleration, because the weight forces are directly
proportional to the mass: P = mg ; a = P/m = mg/m = g .

In the experimental situation considered previously, in which a negative electric charge
is moving inside the uniform electric field produced by a plane capacitor, toward the
positive conductor plane, the electrostatic potential energy of the charge diminishes,
while its kinetic energy increases, and in every point of the trajectory is constant the
sum of the kinetic energy and of the electrostatic potential energy.

This happens, in an analogous mode, in the case of a body that is moving in the Earth
gravitational field passing from a greater height to another one, greater or smaller, and
respectively losing kinetic energy and acquiring gravitational potential energy, or
viceversa, acquiring kinetic energy and losing gravitational potential energy.

The fact that the sum of the kinetic energy and the potential one maintains constant
during the motion, both in the case of a body subjected to an electric field and in the
case of a body subjected to a gravitational field, it expresses the total mechanical
energy conservation principle, applicable to every conservative field.

We observe that happens a continuous conversion of the motion energy ( kinetic energy )
into the position energy (the gravitational potential energy or the electrostatic
potential energy ):

K1 + q V1 = K2 + q V2 = constant.