1.Introduction

The inverse square law arises in the Coulomb’s law. This states that the electric field r due to the charge qlocated at origin is given by

Recently, some measurements were accomplished for the accuracy of inverse square law for Coulomb’s law by considering the small deviation of the inverse square law of the form

where σcan be regarded as a deviation from the inverse square law [¹-³]. Plimpton and Lawton [¹] charged an outer sphere with a slowly varying alternating current and detected the potential difference between the inner and outer spheres. They found σ = 2x 10^-9. In 1970, Bartlett, Goldhagen, and Phillips [²] also achieved an upper limit of σ=1.3×10-13. One year later, Williams and Faller [³] estimated σ=(2.7±3.1)×10-16. From these experimental data for σ, we know that the deviation from the inverse square law is sufficiently small but can be considered to be non-zero.

As a theoretical background of the Eq. (2), we will invoke Wigner deformation. In 1950, Wigner [⁴] proposed a new deformed Heisenberg algebra of the form

where Ris the reflection operator obeying

and we set ℏ=1. It is well known that algebra (3) gives the same equation of motion for the harmonic potential, which implies that algebra is the general second quantization scheme for the classical harmonic potential problem. The coordinate representation for the Eq. (3) is then given by

where Dunkl derivative (DD) [⁵] Dxis given by

and the reflection operator obeys

It is well known that Wigner algebra (3) is linked to the two-particle Calogero model [⁶-⁸] when the Wigner parameter is related to the Calogero coupling constant. Wigner Hamiltonian, which comes from Wigner algebra, has the potential which consists of a simple harmonic potential and an additional inverse square potential. This potential was shown to possess the conformal symmetry and has been the subject of much interest [⁹-¹⁴]. Especially, Macfarlane [¹⁴] found the relation between Wigner algebra and para-statistics.

Here we have a fundamental question. If Coulomb’s law does not obey the inverse square law, what will happen? Is it possible to construct the electrostatics in such case? The answer is YES, which is the main purpose of this work. In this paper, we use DD to derive the deviation of the inverse square law. We apply the deviation of the inverse square law to the electrostatics. This paper is organized as follows. In Sec. 2, we discuss the deformed electrostatics with DD. In Sec. 3, we discuss the modification of DD and deformed electrostatics when the point charge is not located at origin. In Sec. 4, we discuss the deformed electrostatics with a multi-source and continuous source. In Sec. 5, we discuss some examples of electrostatics with discrete charges. In Sec. 6, we discuss some examples of Dunkl-deformed electrostatics with continuous charge distribution.

2.Deformed electrostatics with DD

In this section, we consider the electrostatics with DD. To do so, we introduce the vector DD’s as

where

and

We consider the electromagnetic theory in which the vector differential operator ∇is replaced with the vector DD operator D. Outside the point charge qlocated at the origin, the electric field is assumed to obey the deformed Gauss law outside source

Now let us solve the Eq. (11) by setting

where

Inserting the Eq. (12) into the Eq. (11) we get

Solving the Eq. (14) we get

Thus, the deformed electric field reads

which reveals the law of inverse square with deviation. One can easily check that the deformed electric field obeys

which holds because P×E=0. Thus, the electrostatic potential is defined through

Now let us derive the deformed Gauss law. We assume that the Gauss law for a point charge located at the origin takes the form

where μ(r)is the weight function. The Eq. (19) can be written as

or

Inserting the Eq. (16) into the Eq. (21) we get

which gives

Thus, the deformed Gauss law reads

where ρdenotes the charge density, and for a point charge it is

If we locate a charge Qon r, it will be subject to the force of the form

Now let us find the potential energy for the deformed Coulomb force (26). The Eq. (26) indicates that the deformed Coulomb force is also conservative. Thus, the potential energy is defined through

Solving the Eq. (27) we get

Similarly, the electrostatic potential Vis defined through

which gives

Thus, the electrostatic potential obeys the Dunkl-Laplace equation

3.Modification of DD and deformed electrostatics when the point charge is not located at the origin

In ordinary electro-magnetics, consider the Coulomb force. Let F denote the force acting on a electrically charged particle, with charge qlocated at r, due to the presence of a charge q'located at r'. According to Coulomb’s law, this force is, in a vacuum, given by the expression

The vector DD defined in the Eq. (8) does not obey the above property. Thus, we need a new definition for vector DD when we consider two points, source point, and field point, and this derivative is acted on the function in r - r'. Here we define the DD gradient of the function f(|r-r'|)as

where

and

and the exchange operator obeys

Similarly, we define the DD divergence, and DD curl acted on the vector function of the form f(|r-r'|)(r-r')as

and

The electric field at $\ra$ due to charge q'located at r' is then given by

which satisfies

The deformed electrostatic potential is given by

which gives

4.Deformed electrostatics with multi source and continuous source

Now let us consider the multi source case. Consider the situation that there are Ncharges q1,q2,⋯,qN, each of which is located at r1,r2,⋯,rN. The electric field at r due to Ncharges is defined as the sum of Nsub electric fields as follows:

where the sub electric field due to charge qjis

The sub electric field obeys

and

The electrostatic potential is defined through

which gives

If the discrete electric charges are small and numerous enough, we introduce the electric charge density ρlocated at r' within a volume V'of a limited extent and replace summation with integration over this volume. This allows us to describe the total electric field as integral of the infinitesimal electric field as follows:

where the infinitesimal electric field due to the infinitesimal charge located at r' is

The infinitesimal electric filed obeys

and

The electrostatic potential is defined through

which gives

5.Some examples of Dunkl-deformed electrostatics with discrete charges

Now let us discuss some examples of Dunkl-deformed electrostatics with discrete charges.

6.Some examples of Dunkl-deformed electrostatics with continuous charge distribution

Let us discuss some examples of Dunkl-deformed electrostatics with continuous charge distribution.

7.Conclusion

According to Cavendish type experiments [¹⁵], let us consider that the radii of the two concentric spheres are R1and R2, (R1<R2), with the charges Q1and Q2spread uniformly over them, respectively. Then, the potential at r [¹⁶] is given by

where ris the distance from the origin ( center of concentric spheres). According to Ref. [¹⁷], we get the potential on the inner shell and outer shell as

where

and U(r)is the electrostatic potential for a unit charge. For the ordinary electrostatics, we have U=1/r, which gives f=r. In the deformed electrostatics with DD, from the Eq. (30), we get

For small ν, we have

After the outer shell was charged by a potential V0, a part of the charge would pass through the connecting wire into the inner shell until the equilibrium V(R1)=V(R2)=V0was reached. Then, the charges on the inner shell could be determined as

where

In the ordinary electrostatics ( ν=0), we have Vc(R1)=0, while Vc(R1)≠0for ν≠0. So, by detecting the electrostatic potential on the inner shell directly, one could obtain the deviation from Coulomb’s inverse square law, and then the ordinary derivative in the Maxwell equation could be replaced with DD.

There remains much to be studied in this direction. We should investigate the Maxwell equation with DD and derive the electromagnetic wave equation for the completion of this deformed electro-magnetics. The study of magnetic monopole with deviation seems to be possible with the help of DD. We think that these topics and their related content will be clear soon.