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.

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:

F = q 1 q 2 4 π ε 0 r n = k q A q B r n (4)

including the simple case n = 1:

F = k q A q B r (4a)

Product q_A.q_B from (4), (4a) comprises for simplicity, also coefficients k/4πε₀. 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:

M = F l 2 = m ( q A q B ) (5)

where function m represents the components of total M moments, given by various pairs of q charges.

+ M = m ( q A + q B − ) + m ( q A − q B − ) = k q 2 [ ( 1 r + l / 2 ) . ( l 2 ) + ( 1 r + l / 2 ) . ( l 2 ) ] = k q 2 2 r + l / 2 . l 2 = k q 2 l r + l / 2 (6)

− M = m ( q A + q B + ) + m ( q A − q B + ) = k q 2 [ ( 1 r − l / 2 ) . ( l 2 ) + ( 1 r − l / 2 ) . ( l 2 ) ] = k q 2 2 r − l / 2 . l 2 = k q 2 l r − l / 2 (7)

The ratio r_M of the two moments results from (6) and (7):

r M = + M − M = k q 2 l r + l / r k q 2 l r − l / r = r − l / 2 r + l / 2 < 1 ; → → − M > + M ; (8)

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:

F = − k q A q B r (9)

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:

+ M = m ( q A + q B + ) + m ( q A − q B + ) = k q 2 [ ( − 1 r − l / 2 ) . ( l 2 ) + ( − 1 r − l / 2 ) . ( l 2 ) ] = − k q 2 2 r − l / 2 ⋅ l 2 = − k q 2 l r − l / 2 (10)

− M = m ( q A + q B − ) + m ( q A − q B − ) = k q 2 [ ( − 1 r + l / 2 ) . ( l 2 ) + ( − 1 r + l / 2 ) . ( l 2 ) ] = − k q 2 2 r + l / 2 . l 2 = − k q 2 l r + l / 2 (11)

The ratio r_M of the two moments results:

r M = + M − M = − k q 2 l r + l / r − k q 2 l r − l / r = r − l / 2 r + l / 2 < 1 ; → → − M > + M ; (12)

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:

F = k 1 q A ⋅ q B ⋅ r n (13)

including the simple case n = 1 or the limit power case of lnr, in both cases F force increasing with r:

F = k 1 q A ⋅ q B r             or             F = k 1 q A ⋅ q B ln r (13a),(13b)

Utilizing (13b) we obtain from Figure 1:

+ M = m ( q A + q B − ) + m ( q A − q B − ) = k 1 q 2 [ ( r + l 2 ) ⋅ ( l 2 ) + ( r + l 2 ) ⋅ ( l 2 ) ] = k 1 q 2 2 ( r + l 2 ) ⋅ l 2 = k 1 q 2 ( r + l 2 ) ⋅ l (14)

− M = m ( q A + q B + ) + m ( q A − q B + ) = k 1 q 2 [ ( r − l 2 ) ⋅ ( l 2 ) + ( r − l 2 ) ⋅ ( l 2 ) ] = k 1 q 2 2 ( r − l 2 ) ⋅ l 2 = k 1 q 2 ( r − l 2 ) ⋅ l (15)

The ratio r_M of the two moments results:

r M = + M − M = k 1 q 2 ( r + l 2 ) ⋅ l k 1 q 2 ( r − l 2 ) ⋅ l = r + l 2 r − l 2 > 1 ; → → + M > − M ; (16)

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 is 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.

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_Aq_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 – q A → q A − and having q A − q B + = q A + q B − , 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:

F A = F a 1 A + F a 2 A = k q A − q B + r + k q A + q B − r + 2 l = k q A − q B + 2 r + 2 l r ( r + 2 l ) (17)

F R = F r 1 A + F r 2 A = k q A − q B − r + l + k q A + q B + r + l = 2 k q A − q B − 1 ( r + l ) (18)

If we rewrite the expression from the denominator from (17), one obtains:

r ( r + 2 l ) = ( r + l ) 2 − l 2 (19)

From (19) and admitting r ≫ l it results:

r ( r + 2 l ) < ( r + l ) 2 (20)

Introducing (20) in (17) and comparing with (18) it results:

| F a | = | k q A + q B − 2 r + 2 l r ( r + 2 l ) | > | k q A − q B − 2 r + 2 l ( r + l ) 2 | = | F r | (21)

| F a | > | F r | (22)

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 greater 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).

F = k 1 q A ⋅ q B ln r (23)

with coefficient k₁ 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 a = F a 1 A + F a 2 A = k 1 q A − q B + ln r + k 1 q A + q B − ln ( r + 2 l ) = k 1 q A − q B + ln [ r ( r + 2 l ) ] (24)

F r = F r 1 A + F r 2 A = k 1 q A − q B − ln ( r + l ) + k 1 q A + q B + ln ( r + l ) = k 1 q A − q B − ln ( r + l ) 2 (25)

Comparing F and F_r from Equations (24) and (25), and considering Equation (20) it results:

| F a | = | k 1 q A − q B + ln [ r ( r + 2 l ) ] | < | k 1 q A − q B − ln ( r + l ) 2 | = | F r | (26)

| F a | < | F r | (27)

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'rⁿ 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 about 2 cm radius, which being separated initially, they will tend permanently to gather together as a clew, at a final natural free equilibrium position. Indeed they not separate or move away in space towards an alternate natural equilibrium position in our laboratory scale. Such magnets m_i may constitute a correct intuitive physical model of atomic or nuclear electric dipoles.