## 2 Electrostatics

We shall consider in this Section systems of charges with fixed positions. First, let us briefly review how the concept of Electric Field is formalized.

### 2.1 Coulomb's Law

Coulomb's Law describes the interaction between two charges. It is taken as the customary starting point of Electrostatics, which in turn uses to be the first topic in Electromagnetics.

In terms of forces, Coulomb's Law states that the force upon q' due to q is

Some relevant features are worth to be remarked:

• The force is proportional to the charge, as it may be observed in the following simulation
• Three charges are shown. A central one with Q=+1000 (charge units), and two charges equidistantly placed with respect to the central one, with Q=-20 charge units (left) and Q=+10 charge units (right), respectively. In this way, the interaction between the two charges at the ends is negligible. Notice also that the force between charges of opposite sign is attractive, while between charges of the same sign is repulsive.

• It may also be noticed that, due to its symmetry, Coulomb's Law is consistent with Newton's Third Law, as it is seen in the following simulation
• The two charges considered have the values -20 and +1000, respectively.

• The force is inversely proportional to the square of the distance separating the charges
• In this case, a central charge with value 10000 and a set of charges with values +1 or -1 located at different distances of the first one are shown. The data corresponding to each charge can be visualized in the particle parameter panel by clicking on the charge with the right button of the mouse. The values of force and distance to the central charge can be known in this way.
(At present, the value of 2 for the exponent in Coulomb's Law is known to be exact up to 16 digits. Of course this has not been found from measurements of forces between charges, but from some consequences deduced from Coulomb's Law such as the zero field within a conductor (see the Concept of Metal below).)

• Coulomb's Law is also consistent with the isotropy of space: all directions are equivalent in the absence of charges (and any other physical system). When two charges are placed at given positions a direction is being distinguished from the rest, the one corresponding to the line joining the charges, and this is the direction of the interaction force.
• Validity:
• q' is static (q can move)
• Vacuum and material media
• Relativistic and quantum-mechanical limits of validity
• Electric Field: By putting together all that does not depend on q in Coulomb's Law we arrive at the definition of the field associated to the charge q'
as the force on a unit charge at every point in space.