4.3. The electrostatic force

        When we think of x (electric effects field) as a force field, we should keep in mind that, generally speaking, after the action of x is calculated by trivial methods, it leads us to the force acting on an electron and not on an electric charge. It is also important to observe that the x field of an electron (equation 4.4) is neither of Coulombian nor Gaussian type (the field lines do not begin and finish in the electron). Nevertheless, the x field of a spherical electric charge (equation 4.11) is a Coulomb’s field in its exterior.

        A negative test charge q placed in a uniform x electric field and balanced by its weight will be subject to an electric force F with F = SFiq. Here Fiq is the force exerted by a generic electron i on the electric charge q, as shown in Figure 8.

Figure 8: Comments in the text

        The modulus of F is proportional to the modulus of the x field, or

F = C1x

and thanks to the Superposition Principle valid for forces acting on Coulombian charges we can also write:

        The electric effects field xq produced by the charge q is of the Gaussian type and also of the Coulombian type, according to equation 4.11 for r > R. If xq is not very intense in relationship to x, we can despise the inductive effects. Joining the appropriate proportionality relationships, we obtain:

in which C1, C2 and C3 are constant. Solving the system of equations 4.13 for Fiq, we have:

Fiq = C4xqcosqi

or, in vectorial notation:

Fiq = C4xqcosqik .

        The surface S will be subject to the reaction -F = SFqi. It seems reasonable to expect that Fqi = - Fiq (individualized action and reaction [34]). If we accept this equality, we will get to the expression:

Fqi = - C4xqcosqik .

        It will be convenient to use f to refer to the angle between the w vector electron and the direction of the field to which the electron is submitted. In the specific case under consideration (figure 8) and because the electric field xq is Coulombian this angle equals q, defined through the equation 4.4 (fi = qi). So the previous equation can be written as:

Fqi = - C4xqcosfik .

        To generalize, we will say that the electrostatic force acting upon an electron placed in an electric field x, can be expressed as:

with f = the angle between x and v.