Theoretical assessment of the disparity in the electrostatic forces between two point charges and two conductive spheres of equal radii

The Coulomb's formula for the force FC of electrostatic interaction between two point charges is well known. In reality, however, interactions occur not between point charges, but between charged bodies of certain geometric form, size and physical structure. This leads to deviation of the estimated force FC from the real force F of electrostatic interaction, thus imposing the task to evaluate the disparity. In the present paper the problem is being solved theoretically for two charged conductive spheres of equal radii and arbitrary electric charges. Assessment of the deviation is given as a function of the ratio of the distance R between the spheres centers to the sum of their radii. For the purpose, relations between FC and F derived in a preceding work of ours, are employed to generalize the Coulomb's interactions. At relatively short distances between the spheres, the Coulomb force FC, as estimated to be induced by charges situated at the centers of the spheres, differ significantly from the real force F of interaction between the spheres. In the case of zero and non-zero charge we prove that with increasing the distance between the two spheres, the force F decrease rapidly, virtually to zero values, i.e. it appears to be short-acting force.


Introduction
The formula for the magnitude of Coulomb force Q at a distance R from each other are well known [1], [2]. In reality, interactions take place not between point charges, but between charged bodies of certain geometric form, size and physical structure. It was Maxwell who discovered that electrostatic force between two spheres differed from the electrostatic force between point charges of the same magnitude located at the centres of the spheres [3,Chapter 1]. In his opinion it resulted from the redistribution of charges, due to the mutual electrostatic influence of the spheres. In this way the problem of assessing the deviation of C F from the actual values of the force F of electrostatic interaction arises, especially at relatively short distances.
Using the capacitance coefficients, Lekner [4], [5], [6] investigate the case of spheres at close-approach and derive expression for the force between two spheres. At that, using 2 formula (3.4) from [5], he derives the leading term of the force F. If 12 r r r  and 12 QQ  this term is Here 0.577...

 
is Euler-Mascheroni's constant. Those expressions generalize the force formula for obtained by Kelvin [7]. Khair [8] derive analytical expressions for the electrostatic forces on two almost touching nonspherical conductors held at unequal voltages or carrying dissimilar charges in an insulating medium. Each of those conductors is a body of revolution whose surface is defined by the equation n n n r z a , where r and z are radial and axial cylindrical co-ordinates, respectively, and a and n are parameters that control the particle width and shape.
In the particular case of spheres with 12 r r r  and 12 QQ  for the leading term of the force F he derives formula (17) from [8] Khair notes that his formula is not as accurate as Lekner's.
From formulas (1) and (2) follows that in a close enough distance between the surfaces of the spheres, force F is always a force of attraction.
Using the method of image charges in [9], we were the first to express by infinite sums the exact analytical formula for the force F of electrostatic interaction between two conductive spheres of arbitrary charges Q of the spheres in [9], [10] we have shown already that , where L is dimensionless coefficient. In [11] we have mathematically analyzed the convergence of the coefficient of deviation L.
Here using the formula for L from [9], [10] the possible values of the deviations of C F from F are investigated by means of graphical analysis for different non-zero charges 1 Q and 2 Q as a function of the ratio of the distance R between the spheres to the sum 2r of their radii.  Then from [9], [10] we obtain that force F of electrostatic interaction between two spheres is represented in the form  From these equations, in accord with formulas (7), (9) and (10) from [9], we can derive for the force F : is only expressed through one parameter: the ratio  (10) and (13), it follows that   1 Then, according to the formulas (7) and (11) C FF  . Therefore, equations (7) and (11) appear to be generalizations of Coulomb forces!

Deviation of the Coulomb's force from the force of interaction between two spheres with equal radii.
We will determine the deviation of the Coulomb's force C F and the electrostatic force F between two conducting spheres 1 S and 2 S with equal radii 12 r r r , charged with charges 1 Q and 2 Q . To this end we will discuss the possible values, which it takes, according to the formulas (8) and (13), the respective deviation coefficient   Lk with different values of the real numbers We will study the non-dimensional coefficient    (8) and (13)). It is known, that for nonintersecting spheres 1 2 R r  , i.e. the distance between the surfaces of the spheres 20 Rr  .
Since the spheres' radii are equal, it is sufficient to discuss   Lk only with 11 k    , i.e. when 12 QQ  .
We will discuss the two main cases of charged spheres -with unlike and like charges.
According to formulas (8) and (13)   It could be seen that   1 Lk  for each 10 k    (in the fourth quadrant) and   1 Lk  for each 01 k  (in the first quadrant). From here and from the formulas (7) and (11) for unlike charges the force of attraction between the spheres is greater than the Coulomb force.
We discuss the case   1 Lk  in more detail, i.e. regarding spheres charged with different like charges.
According to formula (8), we present on Fig. 2  This agrees with formula (3.4) from [5], that upon contact of the spheres the force 0 F  . We specify this result as with formula (8)    The deviation between the spheres is smallest when using our exact analytical formula (7) for F. The differences between the obtained values are due to the inaccuracy of the used methods for L F and K F . But when the distances are close enough, the values of F from formula (7) 16 10 QQ

Conclusions and Example
We can see, that with 1 k  is fulfilled 0 C FF  but the values of C F and F do not differ significantly, like we mentioned in the third subcase of (ii). Therefore, with certain accuracy with 12 QQ  we can assume that C FF  .

Discussion
The Coulomb interactions are fundamental in physics. Thus, it is especially important to evaluate the deviation of these idealized interactions from the real electrostatic interactions between conducting bodies. Moreover, this deviation, as determined for spheres in this article, 11 is significant at small distances between such bodies.
Here, we offer an exact method for determination of the deviation of the Coulomb's force C F from the force of interaction F between two conducting spheres with different radii and arbitrary charges 1 Q and 2 Q . For the purpose of finding C F we assume, that 1 Q and 2 Q are located in the centers of the spheres.
With zero 12 QQ  precise the statement of Lekner [5], that at very close distances between the surfaces of the spheres, the force F of interaction between the spheres, is always force of attraction. We find the ratio between the radii and the distance between the centers of the two spheres, charged with like charges, at which the force 0 F  (when the charges are unlike, always 0 F  ). We prove in theory, and specify the experimental results for the interaction of the spheres at relatively close and far distances.
In [12] and [13] numerically are studied the electrostatic interactions, respectively between ellipsoids and tori. The results obtained here could also be used, with approximation, for conducting bodies different than spheres, having center of symmetry, as these bodies are approximated to spheres with equivalent surface areas of the discussed bodies.
Particularly interesting are the cases, when 1 0 Q  , 2 0 Q  and 12 0 QQ  . In the first case we have established that the force F of interaction between the spheres is short-acting, and F is significantly larger than the Coulomb's force 0 C F  . In the second case, it turns out that F is not significantly different than C F and in some studies it could be assumed that C FF  . Similar conclusions could be obtained also for the bond energy W between spheres, in relation to the bond energy C W between point charges. We use these conclusions in [14], [15], [16], [17], as we find the electrostatic interactions between proton-neutron and proton-proton, modeling them as spheres and tori. Using the established experimental data for nucleons, we obtain that the electrostatic forces between proton -neutron are short-acting, and the bond energy is commensurable to the nuclear, while the forces between proton-proton are long-acting and proportionate to the Coulomb's forces.
The results obtained in this article can be used also in intermolecular and other electrostatic interactions.

Conclusion
The results in this work offer a theoretical evaluation of the deviation of the Coulomb interaction C F from the real interaction F between two conducting spheres with equal radii rr . It is established, that if R is the distance between the centers of the spheres with radii 0 r  , then with 21 Rr  the deviation of C F from F increases, and at relatively small distances between the surfaces of the spheres it is significant. In this case the Coulomb interactions 12 cannot be used.
The Coulomb interactions cannot be used also in the case of 1 0 Q  and 2 0 Q  we discus here, in which we find that F is short-acting force.
The article also determines the couple of value If the radii of the spheres are different, then the deviation coefficients will depend on three, and not on two parameters. In this case two of them must be known, in order to analyze the deviation of the forces of interaction C F from F, as well as the bond energies C W from W , depending on the distance between their surfaces of the spheres. This analysis can be done using the general formulas for F and W, which we have obtained [9].