Coulomb discovered the mathematical dependence that governs
the force acting between pairs of point charges. This mathematical
expression is called Coulomb's Law. Coulomb's Law, as any law of physics, is the result of experimental
observations.

In the diagram, represents
the distance between the two charges
and .
The force
is the force on the charge one due to the charge two. Similarly, the force
is the force on the charge two due to the charge one.

Coulomb's law states that the force between the
charges is an action and reaction pair of forces whose magnitude
is proportional to the product of the magnitudes of the two charges and
inversely proportional to the distance square between the charges.
The direction of the forces are along the line going from one
charge to the other. These forces point toward the charges when the
forces are attractive (opposite charges) and the forces "pull" away the
charges when the forces are repulsive (same type of charge).

where is called Coulomb 's
constant and
represents
the magnitude of the Coulomb's force between the charges. The arrows on
top of a symbol indicate that the quantity is a vector and; therefore,
the magnitude and direction of this quantity are needed for their
full identification.

In the case of the top diagram, one of
the charges is positive (red) and the other is
negative (blue).
Therefore, the forces between the charges are attractive. The bottom
diagram shows two charges with the same type of charge; in which case,
the forces are repulsive.

As with any action and reaction
pair, the magnitudes of the two forces are the same; the directions of
the forces are opposite to each other; and, the two different forces act
on different objects (charges). Therefore, on each individual charge,
there is a net force acting on it that will make the charge move along
the direction pointed by the force. In the electrostatic case, it is
assumed that the charges are somehow "nailed" to their location by
forces other than electric forces.

The superposition principle for the forces among electric
charges expresses that if more than two charges are present, the resultant force
acting on one of the charges is the result of the vector addition of the
individual Coulomb forces exerted on this charge by all the other charges.

The drawing sequence on the left illustrates the use
of the superposition principle in electrostatic. The first drawing shows
an arbitrary charge distribution . All these charges will interact
electrically with each other. Nevertheless, to analyze the problem, the
study focuses the attention on one of the individual charges (later the
study could focus on one of the other charges). In this case, the fourth
charge has been individualized for the study. The second drawing,
,
emphasizes this by showing the line of action of the forces exerted by
the other charges on
.
The third drawing shows the actual vectors representing the forces acting
on this charge
. The size of this vector is related to the magnitude of
the Coulomb's force between the individual charges and
.
For the purpose of these drawings, those vector magnitudes have been chosen
arbitrarily and without justification. The fourth drawing,
, shows those
vectors added using the graphical method of adding vectors by connecting
"tip to tails". In addition, the fifth drawing shows the resultant of
such vector addition
. The final drawing in the sequence shows the original
charge distribution with the vector representing the net force acting on
the charge
.
Thus, the sequence represents the graphical addition of

.

The same procedure can be repeated in order to obtain the net force
acting on one of the other charges.

Accordingly with the previous sequence of drawings, in order
to obtain the net force on one of the charges (), the following steps should be
followed:

Use Coulomb's law to calculate the magnitudes of the
individual forces.

Based on the signs of the charges, determine the
direction of the individual forces.

Calculate the net force acting on the charge using the
vector addition of the individual forces.

How should we modify the previous result if the charge
is
changed? What calculations need to be re-done?

The following questions can help to understand the dependency of
Coulomb's law on the different variables.

In reference to the Figure, if d_{2} is twice d_{1}, and q_{1} is the same as q_{4},
the magnitude of the forces exerted by the charge q_{0} over the charges
q_{1} and q_{4} compare as

a)

the force in q_{4}
is twice the force on q_{1}.

b)

the force in q_{4}
is half the force on q_{1}.

c)

the same for both

N

d)

the force in q_{4}
is one-fourth the force on q_{1}.

In reference with the Figure,
if d_{2} is twice d_{1}, and q_{1} is the same as q_{4},
for the forces exerted by q_{0}
on both q_{2} and q_{3} to be the same

a)

the value of the
charge q_{3} must be half the value of the charge q_{2}.

b)

the value of the
charge q_{3} must be twice the value of the charge q_{2}.

c)

the value of the
charge q_{3} must be one fourth of the value of the charge q_{2}.

N

d)

the value of the
charge q_{3} must be four times the value of the charge q_{2}.

In reference to the Figure,
if d_{2} is twice d_{1},
and all the charges have the same value, the
force exerted by q_{0} in the individual charges is the same for

With reference to the Figure,
if d_{2} is twice d_{1},
and if individually q_{1}, q_{2},
q_{3} are the same as q_{4}; then, the forces exerted by q_{0}
on the other charges are

a)

the same for all of
them

b)

greater for q_{2}
and q_{3} than for q_{1} and q_{4}

c)

smaller for q_{1}
and q_{4} than for q_{2} and q_{3}

N

d)

greater for q_{1}
and q_{2} than for q_{3} and q_{4}