6. The electron equation

6.1. The A field
6.2. The fundamental equations of electromagnetostatics
6.3. The scalar electromagnetic field
6.4. The electromagnetic information (e.m.i.)

6.1. The A field

        How does an elementary particle read a field, thus reacting to its implicit content? Let us think of a force field that be electric, magnetic or gravitational. These are effect fields, i.e., areas of space endowed with some particularities and producing measurable effects on certain test objects. Therefore it exists a local and specific potential which is provoked by a physical property continually varying from point to point. Any particle of finite dimensions so long as it is sensitive to this local property reacts to this field; in a similar way to a macroscopic object when submitted to a windstorm.

        An electric field x can be thought of as a specific derivative of another vectorial field A, since A mathematically comprehends this function. So let us suppose that this A field does exist and possesses on every point of its domain all the directional derivatives and is consequently endowed with regular field lines. For example, in order to satisfy hypothesis 4 (H-4), we had to imagine what the A field of an electron should be:

in which represents the derivative according to the field lines of A in relationship to the arch length l or simply the curvilinear derivative according to the l field lines of A in the considered point. It then elapses from equation 5.1 that

and this equation according to the Cartesian reference model (v k) changes into

 

Consequently, Ax = C1 and Ay = C2 with C1 and C2 constant. If we recall that A represents the local content of electromagnetic information in transit emitted by the electron (H-2 and H-3) we must have

and, therefore, C1 = C2 = 0. Under these conditions l coincides with the z axis and the expression for Az has as a solution

Az = K/r + C3 ,

 

which for similar reasons to what was discussed above (for C1 and C2) has C3 = 0. The solution for 6.2 will be with w defined in H-1:

 

agreeing with H-3 and as we shall see below with H-4; therefore, it is the electron at rest equation.

        Now calculating the translational and the rotational of A with its value defined by 6.3 we easily get to the expressions:

 

in which b (in 6.4) satisfies Biot-Savart’s law; therefore, it is the magnetic effect field of an electron at rest and A totally satisfies the basic hypotheses.

        It is interesting to notice that we began our arguments using Coulomb’s law for a sphere with an infinite radius, and we arrived at Biot-Savart’s law; it is more interesting to verify the mathematical reversibility of theoretical path: it is possible to leave from Biot-Savart’s law and arrive at Coulomb’s law. In fact, these laws are mathematically equivalent and equations 6.3 and 6.4 operate the transformation from one into another.