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Notes to a video lecture on http://www.unizor.com

Coulomb's Law

As we know, excess of electrons above the number of protons in an object is what we call negative charge. Deficiency of electrons is called positive charge.
We also know that two positively charged objects or two negatively charged objects repel each other, if positioned close to each other, while oppositely charged (one positive and another negative) attract each other.

It means that there are some forces around electrically charged objects that act in space around them. That is precisely what the term force field means. So, there is an electrical force field that surrounds each electrically charged object.

The next task is to determine the strength of the electrical forces in the electrical field.
Intuitively, the force of attraction or repelling between two electrically charged objects must depend on the number of excessive or deficient electrons in each. The most obvious hypothesis is that the force must be proportional to the number of excess or deficient electrons in each object.

Just as a thought experiment, imagine two point objects A and B with one excess electron in each e1_A and e1_B. The object A will repel object B with some strength F. If the number of excess electrons in object A is increased to two (e1_A and e2_A), the forces of repelling B must be added: one part from e1_A to e1_B and another from e2_A to e1_B. So, the repelling force acting on B will be doubled.

Similar arguments used M times for M excess electrons in object A will lead to multiplication of the initial force by M. If we increase the number of excess electrons in object B to N, we will have to multiply our force by N.
As a result, the total force will be proportional to a product M·N.

We know that the unit of electric charge coulomb is proportional to a charge of one electron. More precisely, the charge of 1 electron is 1.602176634·10⁻¹⁹C.
Therefore, if objects A and B have electric charge q_A and q_B in coulombs, the force of attraction or repelling between them is proportional to q_A·q_B.

The next variable that should be taken into consideration when examining the force of electric field is the distance between objects. The logic we will use to analyze the dependency of the force on distance is similar to the one we used in case of gravitational field.

Consider a set of tiny springs attached to each electron of the charged object A. Each such spring represents a force developed by one electron.
Obviously, the greater the distance between a probe object B and object A - the smaller is the density of springs in the space. It is intuitively obvious that the force of attraction or repelling acting on object B is proportional to a density of springs in the area where B is located, the less springs are observed where B is - the smaller the force will be and vice versa.

In turn, the density of springs is inversely proportional to a square of a distance from object A because the area of a sphere equals to 4πR², where R is a radius.

Therefore, the force of an electrical field at a distance R from its source (a charged object) is inversely proportional to a square of R.

Summarizing all the above, we suspect that the force of attraction or repelling between two charged objects with charges q_A and q_B at a distance R from each other should be proportional to
q_A·q_B / R²

Experimentally, this was confirmed and the only detail we need is to adjust the units of measurement, so the resulting force will be in the units we usually use.
In the SI system, where the force is expressed in newtons, electric charge in coulombs and the distance in meters the force of attraction or repelling is
F = k·q_A·q_B / R²
where
F is the magnitude of the force of attraction (in case of opposite charges) or repelling (in case of the same type of charge, positive or negative) in N - newtons
q_A is electric charge of object A in C - coulombs
q_B is electric charge of object B in C - coulombs
R is the distance between charged objects in m - meters
k is a coefficient of proportionality, equals to 9.0·10⁹ in N·m²/C²

The above is the Coulomb's Law, discovered by French physicist Charles-Augustin de Coulomb in 1785.

The Coulomb's Law describes the force between two charged objects. If both have the same "sign", both are positively charged with deficiency of electrons or both are negatively charged with excess of electrons, the sign of the force is positive, it's a repulsive force. If the charges are of opposite "sign", one positive with deficiency of electrons and another negative with excess of electrons, the force is negative, it's attractive force.
The word "sign" we took in quotes because it's just an artificial way of designating different types of charges that physicists use for convenience.

As we see, the Coulomb's Law for electric field looks very similar to the Newton's Law for gravitational field.
The fundamental difference between these two fields is that gravity always attracts, while electrically charged objects can attract or repel each other, depending on what kind of electrical charge they have. This is the asymmetry of gravitation and a strong argument to consider gravitational field as something fundamentally different from electrical field. Indeed, the General Theory of Relativity by Einstein suggests that the gravitational field is the result of curvature of the space we live in.

Another purely quantitative difference between these two fields is the magnitude of the force.
Consider, hypothetically, two electrons at a distance of one millimeter from each other. They are repelling each other because of electric force, that depends on their charge, and attract each other because of gravity, that depends on their masses.
Let's compare these two forces using the Coulomb's Law and the Newton's Law, using standard SI units.
Electric charge of an electron is 1.602176634·10⁻¹⁹ C.
Mass of an electron is 9.1093837015·10⁻³¹ kg.

Electrical force
(repelling)
F_e = k·q_A·q_B / R², where
k=9.0·10⁹
q_A=q_B=1.602176634·10⁻¹⁹
R=0.001
Resulting electrical repelling force is
F_e ≅ 2.31·10⁻²² N

Gravitational force
(attracting)
F_g = G·m_A·m_B / R², where
G=6.674·10⁻¹¹
m_A=m_B=9.1093837015·10⁻³¹
R=0.001
Resulting gravitational attracting force is
F_g ≅ 5.53·10⁻⁶⁵ N

As you see, the difference is huge. Electrical force between two electrons is significantly stronger that the gravitational force. On a subatomic level the gravitational forces can be ignored. On a planetary level the electrical charges are often small, planets are, generally speaking, electrically neutral or very close to neutral, so the gravitational forces play the major role.