Analysis of Electrical Dipoles Interaction Forces as a Function of the Distance and of the Form of Electrical Force Law

Here, we initially introduced and demonstrated two principles: orientation OR principle and attraction AT principle of electrical dipoles. The OR principle stipulates that any two electrical dipoles P1A, P1B, from two bodies A and B, at any distance in the free state each, will be reciprocally oriented parallel and in the same sense if the electrical interaction forces F between them are of decreasing type with distance r. If the electrical interaction forces F are of increasing type with distance, the two dipoles will be reciprocally oriented parallel but on the opposite sense. The AT principle stipulate that any two electrical dipoles P1A, P1B, at any distance in the free state each, will present always a reciprocal force of attraction FD in both cases of orientation accordingly to OR principle in case of any type of electrical force F decreasing or increasing with distance. These findings may complete our previous work where we found that FD force, between two electrical dipoles P1A, P1B considered at atomic and nuclear level, is in fact the actual gravitation Newton force FN. The paper must be considered together with this work for more consistency.

planetary scale and even at building scales, the electrical forces F C given by classical Coulomb's law of electrical forces: 1 2 1 2 2 2 0 4π C q q kq q F r r ε = = ; with k = 1/4πε 0 ; (1), (1') between two neutral bodies are negligible and should not be taken into account, as it happens today.
The only forces considered today at such long distances are the gravitational forces F N given by classical Newton's law.
This affirmation excludes the electromagnetic phenomena which consist of more complex combinations of electric and magnetic forces originated at atomic scales.
But it is unlikely that the interaction forces acting between two electrical charges q 1 , q 2 may be perfectly described by mathematical simple laws as in (1), at any distances r, and of any scales, in the micro and macro universe.
At this point of our analysis, the above affirmation is justified by the reality complexity or its infinity and also by discrepancies between calculus results of actual simple laws from (1), ( 2) and some experimental measurements and observations concerning these forces, reported in the past.
Discrepancies concerning the Coulomb's law are confirmed by modern experiments, which allow scientists to conclude that in the case of very short distances (such as between molecules, atoms, nuclei, particles) the law (1) is not precisely correct [1], but also it has not yet been proven in the case of long distances [2].In such circumstances, we can conclude that the Coulomb's law in the current form (1) is not perfect and that it can be improved.
For a more accurate description of the infinite reality, including ether presence which will be considered throughout this paper, as we previously demonstrated [3] [4] [5] [6], the most appropriate (but not singular mathematical) mode of electrical forces description, will be probable, a long or infinite series of successive powers of the 1/r distance.
In the present, it is accepted that macroscopic bodies have a random distribution of +q and -q charges, which will generate a random distribution of dipoles in any direction, and all coefficients k i from (3) are canceled [7].
This means that starting from small distances from any neutral body to infinite distances, the influence of the electric charges from that body is considered today to be null.But we will show below that this approach isn't correctly justified.

Justification of Introducing the Two New Principles of Electric Dipoles Interaction, OR and AT
Despite the fact that bodies A, B…, are neutral, the +q and -q charges are in fact always separated in space at atomic and nuclear scale, and consequently every charge +q or -q will exert its electrical interaction force F C to infinite distance according to (1) or to a corrected Coulomb law as in [8] [9] [10] [11].
So in nature practically all the matter must be regarded as being organized always as electrical dipoles P1, P2, P3… as result of Coulomb law action even at smallest scales.
We will show next, that neglecting electrical forces and their potential even for distant bodies is not correct because we can admit and demonstrate the following two principles: a) OR principle of the reciprocal orientation of distant dipoles.
b) AT principle of the permanent attraction of oriented distant dipoles.

Presentation and Justification of OR Principle
In order to demonstrate the OR principle, we can calculate the torsion moments +M (clockwise) and −M (counterclockwise) created by an oriented dipole P2 from body B upon an un-oriented dipole P1 (disposed at 90˚ in body A).
In this situation (Figure 1) we can distinguish two cases, depending on the variation mode with distance r, of interaction Coulomb's forces F (reference frame xOy is attached to A body).

Case 1a
). of decreasing F forces.Firstly, for this calculus, we suppose the interaction forces F a (attraction for +q/−q charges when F a < 0, according to (1)) and F r (rejection for +q/+q or -q/−q charges, F r > 0, according to (1)) between electrical charges +q and -q as being some vectors F of central orientation decreasing with the increasing distance r l  , between body A and body B according to the law of the general type (4), generalizing (1), with n a natural number: including the simple case n = 1: Product q A .q B from ( 4), (4a) comprises for simplicity, also coefficients k/4πε 0 .
Utilizing (4a), we obtained from Figure 1 the clockwise torsion moment +M and counterclockwise −M moment, given upon P1 dipole, by forces F from P2 dipole, as follows: ( ) where function m represents the components of total M moments, given by various pairs of q charges.

. 2
A B A B l l M m q q m q q kq r l r l l l kq kq r l r l

. 2
A B A B l l M m q q m q q kq r l r l l l kq kq r l r l The ratio r M of the two moments results from ( 6) and ( 7): This result of M moments ratio r M indicates that the orientation of dipole P1 will be in the same sense/direction as dipole P2, in this case of the decreasing positive forces F.
Case 1b). of decreasing F forces and sign change.We now must calculate what happens in the hypothetical case when the forces F from ( 4) and (4a) will change the sign from positive +F to negative −F forces, including the simple case n = 1: In this modified Case 1b), we admit hypothetically, an inversion of the actual physical rule (attraction force -F a between +q and -q charges), becoming repulsion force +F r between +q and -q charges and attraction force -F a between +q and +q or between -q and -q charges.
In this modified Case 1b) the Equations ( 6) and ( 7) becomes, considering the same senses of rotation of +M and -M moments, as in Figure 1:

. 2
A B A B l l M m q q m q q kq r l r l l l kq kq r l r l The ratio r M of the two moments results: This result of the M moments ratio r M indicates that the orientation of dipole P1 is in the same sense/direction as dipole P2, also in the case of inverse sign of decreasing forces F (as in Case 1a). of decreasing actual forces F).
So we conclude that changing the sign of F force, have no influence upon the orientation rule of dipoles P1 and P2.
Case 2a). of increasing F forces.Secondly, for this calculus, we suppose the interaction force F a (attraction) and F r (rejection) between electrical charges q as being some vectors of central orientation increasing with the distance r l  , according to the laws of the general type (13), with n a natural number: including the simple case n = 1 or the limit power case of lnr, in both cases F force increasing with r: or ln Utilizing (13b) we obtain from Figure 1: The ratio r M of the two moments results: This result of M moments ratio r M indicates that the orientation of dipole P1 is in the opposite sense/direction as dipole P2, in case of increasing positive forces.
In Case 2b), of increasing F forces and sign change, when the force F is also as increasing with r type, but of reversed sign, is similar to the Case 1b).And hence the result of the calculus of r M ratio will be similar: indicating that the orientation of dipole P1 is in the opposite sense/direction as dipole P2. (as in Case 1b). in this case of increasing positive forces F.
And so at points Case 1) and Case 2), the OR principle was demonstrated.
The OR principle stipulate that in case of decreasing electrical forces F with r the orientation of dipole P1 in the same sense/direction as dipole P2 irrespective of sign of F.
But in case of increasing electrical forces F with r, the orientation of dipole P1 is in the opposite sense/direction as dipole P2 irrespective of sign of F.

Presentation and Justification of AT Principle
In order to demonstrate the AT principle, will be calculated forwards the resultant force R, appearing between two oriented dipoles, P1A and P1B, according to OR principle, given by forces of attraction F a and by forces of repulsion F r existing between +q and -q charges, as in actual F C or completed F, Coulomb's law.
It must determine if the force R is the attraction or repulsion force, as a function of the form of the electric force law F.
The situation of two dipoles P1A and P1B originated in A body and in B body respectively, identically oriented and equal to each other, with polar moment value r = q•l, is presented in Figure 2. The two possible cases of the F force variation with the distance r between two electric charges will be analyzed below: Case 1) the decreasing variation and Case 2) the increasing variation of F force with r.
But we must remind that first calculus concerning the attraction force F a and the rejection force F r between two farther electrical dipoles was performed by us in [8] [11], but without presenting there the significance of the result of that calculus, consisting in the sign of resultant force R. Also there in [8] [11], was made a simplification of calculus by introducing a substitution: α = kq A q B .But here we will emphases that such result, will conduct at a new important principle, the AT principle of permanent attraction of electric dipole, described firstly here below.
In order to better understand the origin of AT principle we will resume here without the above substitution, the calculus of forces F a and of F r from [8] [11] performed in the two cases, Case 1 and Case 2, of the F force variation with the distance r.Case 1). of decreasing F forces.Firstly, we suppose the interaction force F a (attraction) and F r (rejection) between electrical charges +q and -q as being some vectors of central orientation decreasing with the distance r l  , according to the laws of the general type, from Equations ( 4) or (4a).From Figure 2, we now can calculate directly by simple algebra, utilizing Equation (4b), the total force upon each dipole, exerted by the other dipole.Here in calculus, we consider the + sign for force values having the sense of real force as in Figure 2, which corresponds to the attraction force F a (F a < 0) and to repulsion force F r (F r > 0), forces exerted by the charges +q, −q, as in Coulomb law (4).
From Figure 2 notating -A A q q − → and having A B A B q q q q − + + − = , we consider first, the forces exerted upon P1A dipole by a P1B dipole.Utilizing the simple case (4b) the resultant forces F a and F r values results as follows: ( ) kq q kq q r l F F F kq q r r l r r l ( ) If we rewrite the expression from the denominator from (17), one obtains: From ( 19) and admitting r l  it results: Introducing (20) in (17) and comparing with (18) it results: r l r l F kq q kq q F r r l r l From ( 22) it results that in the case of forces F of the type as in Equations ( 4), (4b) decreasing (n = +1) with the r distance, the attraction force F a will be great-er than the repulsion force F r and finally an attraction force R a = F a − F r > 0 between the two identically oriented dipoles P1A, P1B results.This result is correct even if the power of r is any n > 1, as can easily be demonstrated.
Case 2) of increasing F forces.Let's suppose the force F increases with the distance r, including a law of the form (13b).
with coefficient k 1 being similar with coefficient k from Equation (4a) but having the appropriate measure units considering that the term lnr must be adimensional.
Again from Figure 2, for identically oriented dipoles P1A, P1B (Figure 2), the total forces upon P1A dipole utilizing (23) will be: F F F k q q r k q q r l k q q r r l Comparing F and F r from Equations ( 24) and (25), and considering Equation (20) it results: From ( 27) it results that in the case of increasing F forces of the type from (23), the repulsion force F r between two identically oriented dipoles P1A, P1B, will be greater than the attractive force F a and the resultant force R r = F r − F a > 0, will be a repulsion force.This result is correct even if the increasing variation has the general form F = α'r n with n > 1, as it can easily be demonstrated.
But in this Case 2 of increasing F forces (23), the natural orientation of two dipoles becomes reverted compared with those from Figure 2, according to the OR principle from Section 2.2., and the interaction force between the two dipoles P1A and P1B will be again an attraction force R a = F r − F a < 0 as for decreasing forces.So, the AT principle stipulate that in case of decreasing electrical forces F with r an attraction force R a between the two identically naturally oriented dipoles P1A, P1B results.
Also in case of increasing electrical forces F with r, because the two dipoles P1A, P1B will be naturally inverse oriented accordingly to OR principle, AT principle stipulate that again an attraction force R a between the two dipoles P1A, P1B results.
And so, the simultaneous action of OR principle and of AT principle, will give birth of a permanent attraction between two any electrical dipoles irrespective of F force type.
The above analytical calculus and its results, yields the above OR principle and AT principle, whose result must be real.This effect of permanent attraction of electrical dipoles P1A, P1B can be observed also in case of a group in any number of usual magnets m i as spheres of

Figure 1 .
Figure 1.Situation of two distant dipoles P1 and P2 from bodies A and B, which manifest a reciprocal orientation with xOy frame attached to A body.

Figure 2 .
Figure 2. Forces F acting between a pair of identical oriented dipoles P1A and P1B considered at atomic and nuclear level, from A and B bodies with xOy frame attached to A body.